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Star Products that can not be induced by Drinfel'd Twists

This is the final version of my master thesis. I prove that there are no twist star products on the 2-sphere and on the higher genus pretzel surfaces deforming a symplectic structure. One of the key arguments is that a connected compact symplectic manifold endowed with a twist star product has to be a homogeneous space. Furthermore, there is a non-degenerate r-matrix on a Lie algebra whose Lie group is acting transitively and effectively on the manifold in this case. I recap the theory of homogeneous spaces, in particular the classification of transitive effective actions on the 2-sphere, and discuss the notion of r-matrices and Drinfel'd twists. Some results of this thesis are available as a preprint with Pierre Bieliavsky, Chiara Esposito and Stefan Waldmann on arxiv.org under the title"Obstructions for Twist Star Products".


Summary of this Thesis
We give obstructions to the existence of Drinfel'd twists on connected compact 2-dimensional symplectic manifolds. In fact, only the 2-torus permits a twist star product that deforms the symplectic structure. The main observation in the line of arguments is that a twist star product forces such a symplectic manifold to be a homogeneous space. This immediately excludes the higher pretzel surfaces T(g) since they can not be structured as homogeneous spaces: by a theorem due to G. D. Mostow the Euler characteristic χ of a compact homogeneous space is non-negative, while χ(T(g)) = 2 − 2g. The 2-sphere S 2 is a bit more involved. A. L. Onishchik classified all connected Lie groups that act transitively and effectively on S 2 up to equivalence.
In particular, they are semisimple Lie groups. We prove that a twist star product on S 2 induces a transitive effective action of a connected Lie group on S 2 that is not semisimple, to produce an obstruction also in this situation. It is the Etingof-Schiffmann subgroup of the r-matrix corresponding to the twist.

Deformation Quantization
The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.

-Eugene Wigner, The Unreasonable Effectiveness of Mathematics in the Natural Sciences
A fundamental strategy in applied science is approximating physical systems by mathematical formalisms. The advantage of this idea is that mathematical implications can lead to predictions on the corresponding physical system. However, one has to be careful: a mathematical theory is just an approach of nature. If it is in conflict with a reproducible measurement, the mathematical framework has to be adapted. At its best by a more general theory which includes the aspects of the old theory that are conform to the physical results. One might think of quantum mechanics extending classical mechanics. Even if the laws of classical mechanics seem more nearby in daily life, quantum mechanics is said to be the best description of nature today. This statement is based on measurements on very small scales. But there is an open question: how does the classical situation appear as a limit of the quantum mechanical system? Remark that this is a pure mathematical question, since we already know that both, the classical and the quantum world, exist. There are several approaches of quantization, e.g. operator formalism on Hilbert spaces (consider [31,79]) or path integral quantization (see [41,86]). The one we are interested in is the so-called deformation quantization. To explain its essence we give a concrete example following [107, Section 9.1.1].
We want to discuss an approach to describe a particle of mass m influenced by a force field F : R 3 → R 3 . The position of this particle at the time t ∈ R is represented by a vector q(t) ∈ R 3 as well as the momentum p(t) = mq(t) of the particle. According to Newton's second law one has F (q(t)) = mq(t), (1.0.1) for any time t ∈ R, whereq(t) denotes the acceleration of the particle at the time t. This second order differential equation is equivalent to the two first order differential equationṡ q(t) = 1 m p(t) andṗ(t) = F (q(t)). (1.0.2) Remark that we made a change of coordinates from q(t) ∈ R 3 to (q(t), p(t)) ∈ R 6 . If we assume F to be conservative there is a potential V : R 3 → R such that and combined coordinates x = (q, p), we are able to reduce Hamilton's equations of motion to a single first order vector-valued differential equatioṅ x(t) = Ω((∇H)(x(t))). for all f, g, h ∈ C ∞ (R 6 ). The space of smooth real-valued functions on R 6 equipped with the pointwise product of functions is said to be the algebra of classical observables. In deformation quantization one does not change the space of variables by passing to the quantum mechanical system. Instead of this, the multiplication is changed: one deforms the associative and commutative multiplication of the classical observables by adding perturbation terms, such that the result is still associative but not commutative any more. This has to be done in a way such that the classical limit gives back the pointwise product. One also wants the new product to satisfy the correspondence principle: the first perturbation term of the commutator with respect to this product has to result in the Poisson bracket of the classical system. Returning to our example, we define the Weyl-Moyal star product (i ) n C n (f, g), (1.0.14) where C n are bidifferential operators and denotes Planck's constant, the formal parameter of this series. This is the first step of deformation quantization: we obtained a noncommutative algebra (C ∞ (M ) [[ ]], ) of quantum observables. The product is not commutative and we realize that One approach is the GN S construction (see [104]). In a general setting, the classical observables are encoded by the algebra of smooth functions on a Poisson manifold (see Section 3.2 for a definition). Then a star product (see Definition 4.2.3) is a deformation of the pointwise product of these functions, such that the correspondence principle leads to the Poisson bracket of the manifold. Thus, if one is interested in quantization of a classical system, one option is to consider star products on Poisson manifolds. The notion of star products goes back to the work of F. Bayen et al. [9]. S. Gutt [47] and V. G. Drinfel'd [33] discovered star products as deformations of linear Poisson structures on the dual of Lie algebras. Later B. V. Fedosov proved that there is a star product on any symplectic manifold (c.f. [40]) extending the work of M. de Wilde and P. Lecomte (c.f. [29]). The contribution of Fedosov is remarkable, since he gives a recursive formula of a star product, while constructive proofs are rare in deformation quantization. Ideas of his proof are also used beyond symplectic geometry, e.g. on Poisson manifolds (c.f. [26]) and Kähler manifolds (c.f. [19]). By a result of M. Kontsevich (c.f. [61]) there is always a star product on a Poisson manifold. A different proof by D. E. Tamarkin appeared later (c.f. [99]). A physically more relevant approach is to consider convergent star products. This can be looked up e.g. in [17,95]. Consider also [32] for a star product approach to quantum field theory. The papers [97] and [106] overview the development of deformation quantizations and provide further references.

Drinfel'd Twists
We are interested in a more specific situation, again motivated form physics: if there is a quantization of the symmetries of a classical system, the system itself has to be quantized in a compatible way. This led to the notion of twists in the 1980's. It was V. G. Drinfel'd who developed the corresponding theory (see e.g. [33,34]). Symmetries are encoded by formal power series of universal enveloping algebras that act on the algebra of classical observables. The twist is an element of the tensor product of the symmetries. As a result there is a way to change the Hopf algebra structure of the symmetries such that the new structure is noncocommutative. Moreover, one can deform the product of the classical observables and obtains a noncommutative product in terms of the twist. If is a star product, we received a deformation quantization of the classical system via a twist. Thus the advantage is that any module algebra of the symmetries is deformed in an appropriate way. There has been many research on twist deformation: in [35] Drinfel'd proves that twists are in one-to-one correspondence with left-invariant star products on formal power series of smooth functions on Lie groups. Concrete examples and formulas can be found in [11] and [44], while [6] gives an extension of twists to arbitrary connections. Via formality G. Halbout provides a quantization of twists on Lie bialgebras to obtain quantum twists (c.f. [48]). Furthermore, twists enable deformation quantization e.g. for actions of Kählerian Lie groups (c.f. [16]), 3-dimensional solvable Lie groups (c.f. [12]) and the Heisenberg supergroup (c.f. [13]). In quantum field theory one uses twists to deform the Poincaré group and implement the framework of noncommutative geometry (see [59]). There are even twist approaches in string theory (consider [4,5]).
Drinfel'd introduced the twist element  19) provides a deformation of a product ·. But one deforms in a very controlled way: the new product is associative spaces for our purpose. Also three possibilities on how the 2-sphere can be structured as a homogeneous space are given. It was already mentioned that compact homogeneous spaces have non-negative Euler characteristic. We give a proof assuming slightly stronger conditions by using group-invariant de Rham cohomology and Lie group representation theory. This chapter ends by summarizing the classification of connected Lie groups that act transitively and effectively on the 2-sphere. The following chapter is more algebraic. After studying some basic Lie algebra representation theory, we consider Lie bialgebras and mention Poisson-Lie groups as their global counterpart. In particular, the dual character of Lie bialgebras is pointed out. In fact, we are interested in a more specific situation: some Lie bialgebra structures are cocycles, i.e. they are obtained as the Chevalley-Eilenberg differential of an element r ∈ g ∧ g. This naturally leads to r-matrices and the classical Yang-Baxter equation. While until now most of the theory was basic, the last section of this chapter is quite specific: we define the Etingof-Schiffmann subalgebra as the Lie subalgebra, in which a r-matrix is non-degenerate. The Etingof-Schiffmann subalgebra is never semisimple, which is an essential argument. The first section of Chapter 4 is again very algebraic. We examine the notion of Drinfel'd twists in great detail. First we focus on twists on arbitrary Hopf algebras and in the second section on twists on formal power series of the universal enveloping algebra. As mentioned in the introduction, the twist induces deformation in a more categorial frame: not only the Hopf algebra but every module algebra can be deformed. The deformed algebra multiplication is said to be a twist star product in the context of formal power series and star products. As a last statement, we view a twist on formal power series as a quantization of a r-matrix. In Chapter 5 we prove one of the main results of the thesis by connecting all previous notions and results. Assuming the Poisson bivector of a connected compact symplectic manifold to be the image of a r-matrix under a Lie algebra action, we construct a transitive Lie group action on this manifold. This Lie group is the Etingof-Schiffmann subgroup corresponding to the r-matrix. Thus this special Poisson bivector is an indicator for homogeneous spaces. In particular, this situation occurs if there exists a twist star product. Thus we prove that a connected compact symplectic manifold, that can be equipped with a twist star product, is a homogeneous space and there is a r-matrix which is non-degenerate in a Lie algebra, whose Lie group acts transitively on the homogeneous space. In the last chapter we use this result to produce obstructions to twist star products. The higher pretzel surfaces are not homogeneous spaces and the sphere only admits semisimple transitive effective Lie group actions, which stands in contradiction to the action of the Etingof-Schiffmann subgroup. Remark that there are star products on all these symplectic manifolds according to the Fedosov construction. Thus there are indeed star products that can not be induced by Drinfel'd twists. The two appendices are short introductions to Hopf algebras and semisimple Lie algebras, respectively.

I want to thank my advisors, Chiara Esposito and Stefan
Waldmann, for their support and patience. They invested a lot of time in discussions that promoted the thesis. Not to mention the nice bqs regardlessly of the weather. I am grateful to Pierre Bieliavsky, since the idea that led to this thesis is based on a discussion of him with Chiara Esposito and Stefan Waldmann in Oberwolfach [14]. I want to thank Jonas Schnitzer and Thorsten Reichert for giving me their assistance and advice. Finally, I am thankful to my family and Verónica for their encouragement that strengthens me.

Transitive Actions and Homogeneous Spaces
Homogeneous spaces are one of the central concepts of this thesis. A first and very simple definition of a homogeneous space is the following: if H is a closed subgroup of a Lie group G, the set of left cosets of H in G, i.e. {{gh | h ∈ H} | g ∈ G}, is said to be a homogeneous space G/H. This also works for topological groups. But with this point of view we do not see the actual strength of homogeneous spaces. The original motivation to consider them is their nature to encode symmetries of spaces. To make this idea a bit more concrete one needs the notion of Lie group actions. They are smooth functions that assign to any point of a Lie group G a diffeomorphism on a smooth manifold M in a way that respects the left multiplication on G. Also, one wants the identity on G to induce the identity map on M . There a various types of Lie group actions with many nice and useful properties and we discuss the ones that are interesting for our purpose in Section 2.1. For any Lie group action there are two fundamental structures, one on the Lie group and one on the manifold. The structure on the Lie group is the set of elements which induce diffeomorphisms that reproduce a given point x on M and it is said to be the stabilizer of x. We can prove that it is always a Lie subgroup of G. On the other hand the interesting structures on the manifold are the orbits of the action. Intuitively, everyone has an idea of what an orbit should be: one might think of the way a satellite circuits a planet or electrons an atomic nucleus in the Bohr model. There is a way to phrase these ideas also in a mathematical definition: for a fixed point x on the manifold M the orbit of x is the set of points on M that are reached by applying x to every diffeomorphism that is produced by elements of G via the action. Unfortunately, not every orbit is a smooth submanifold of M , but we prove that this is true if one stays close to the identity in G or if the Lie group is compact. A very interesting situation is obtained if there is only one orbit, i.e. if all elements of x produce the same orbit which coincides with M in this case. Such actions are said to be transitive and they connect homogeneous spaces to Lie group actions. It is the task of Section 2.2 to establish this connection. First of all, it is easy to conclude that the left cosets of a closed subgroup H of the Lie group G are exactly the orbits of the right multiplication of elements of H in G. Thus one could define a homogeneous space G/H as the orbit space of this particular action. One benefit of this definition is that one is able to build the structure of a smooth manifold on the homogeneous space as a simple consequence of the "Free and Proper Action Theorem". But there is more: we get a whole classification of homogeneous spaces by transitive actions. On one hand there is a natural transitive action of G on any homogeneous space G/H. Surprisingly, the converse statement is also true: for any transitive action of a Lie group G on a smooth manifold M there is a diffeomorphism between M and the homogeneous space G/G x , where G x denotes the stabilizer of any point x of M . Thus we are able to identify M with the orbit space G/G x , i.e. any point of M is viewed as a left coset or orbit of an element in G with respect to G x . This also explains why one calls G the symmetries of M which was one of our initial statements. The last perception is fundamental for the proceeding of this thesis. Instead of searching for coset or orbit spaces we are looking for transitive actions, that are easy to handle.
As an example, we give three different ways to structure the 2-sphere as a homogeneous space via different Lie groups that act transitively on it. The last two sections of this chapter are then more specific classifications and indicators of homogeneous spaces with topological arguments. Section 2.3 is all about to prove that connected compact homogeneous spaces have non-negative Euler characteristic. The Euler characteristic is a topological invariant of a manifold, namely the alternating series of the dimensions of its cohomology groups. It is an integer that is known for many manifolds. In particular, the connected compact pretzel surfaces T(g) of genus g ∈ N 0 have Euler characteristic 2(1 − g). It follows that T(g) is not a homogeneous space if g > 1 and we experience the first time a very typical spirit of this thesis: we found an obstruction, i.e. there is no Lie group that acts transitively on those manifolds. In Chapter 6 we expand these obstructions to twist star products. But first we continue with Section 2.4 and classify all transitive actions that act on the 2-sphere with some additional properties. In fact, these are the three Lie group actions that we discussed in Section 2.2 before. In particular, they are semisimple Lie groups.

Lie Group Actions
We have to assume that the reader has a basic understanding of Lie theory. The symmetries of our systems are represented by Lie groups and we often investigate their infinitesimal information, i.e. their Lie algebras. For a introduction to these two fundamental objects and their connection we refer to [22,Chapter III], [36,Chapter 1], [63,Chapter 4] or [87,Chapter 4]. Most of the time we argue with a very geometric point of view, thus the proofs often inherit tools known from differential geometry. An efficient approach to this huge topic is given in [68] while we also refer to [69] and [71]. The notion of Lie group actions is also very common, but nevertheless we present the basic definitions and results in this section since these ideas are very central in this thesis. However, we proceed in a colloquial style to accelerate the process, but also to carve out the important concepts. In the following M always denotes a smooth manifold and G a Lie group with identity element e ∈ G. A (left) action of G on M is a smooth map that satisfies Φ(g, Φ(h, x)) = Φ(gh, x) and Φ(e, x) = x, for all g, h ∈ G and x ∈ M . We further denote for any g ∈ G the induced diffeomorphism on M by This is clearly a diffeomorphism since there is the inverse map Φ g −1 and the smoothness of both maps follows from the smoothness of Φ. Thus one can consider which is a contradiction to the effectiveness of Φ. In particular, this situation is fulfilled if for two different elements . This kind of action is said to be free. In general kerΦ ⊆ G is a Lie subgroup. This holds becauseΦ is a group homomorphism by the action property of Φ. If kerΦ is a discrete subgroup we call the action Φ locally effective. Thus by definition any effective action is locally effective but not vice versa. Two more notions induced by actions can be obtained by considering for any x ∈ M the smooth map We call the image Orbits are interesting objects because they build a partition of M , i.e. M can be regarded as the disjoint union of the orbits of any action Φ on M . To prove this one first recognizes that the orbits of two different points x, y ∈ M are either equal or disjoint, since if (G · x) ∩ (G · y) = ∅ there is an z ∈ (G · x) ∩ (G · y), i.e. there are g, h ∈ G such that Φ x (g) = z = Φ y (h). Then for any m ∈ G the point Φ y (m) of the orbit G · y is an element of G · x since In the same fashion G · x is a subset of G · y which implies G · x = G · y. Also any point x ∈ M lies inside one orbit, namely in its own since Φ x (e) = x. Thus we achieved the partition of M as it was stated above. This gives the possibility to define an equivalence relation on M by declaring x ∼ y for x, y ∈ M if and only if x and y lie in the same orbit. We are interested in the situation in which there is only one orbit because such actions are very much related to homogeneous spaces. We call them transitive actions. Equivalently, one can say that an action is transitive if for each pair (x, y) ∈ M × M there is an element g ∈ G such that Φ(g, x) = y.
One missing piece to connect transitive actions to homogeneous spaces is the second object we defined along with orbits, the stabilizer. In the next section we see that in the case of a transitive action all stabilizers are conjugate to each other and the quotient space of the Lie group and any stabilizer is isomorphic to M . This procedure is said to structure M as a homogeneous space via a transitive action. Moreover, it turns out that the concepts of homogeneous spaces and transitive actions are equivalent. Before we study this connection we want to prove some results that already apply to arbitrary actions. The question is what kind of substructures orbits and stabilizers are. For this we consider the map Φ x for a x ∈ M and prove that is has constant rank. Remember that the rank rank p f of a smooth map f : M → N between smooth manifolds at a point p ∈ M is the dimension of the image of the derivative T p f : T p M → T f (p) N of f at p, i.e. rank p f = dim imT p f . If the rank does not depend on the point p which was chosen, the Constant Rank Theorem (c.f. [68,Theorem 8.8]) states that each level set of f is a closed embedded submanifold of M of codimension rank p f . In this case we denote the rank of f by rankf since the point we choose is not relevant. A further consequence of this theorem is that holds for all q ∈ N (c.f. [68,Lemma 8.15]). Also recall that f is a local diffeomorphism if for ii.) The stabilizer G x ⊆ G is a Lie subgroup of codimension k and iii.) There is a neighbourhood U ⊆ G of the identity such that the set U · x ⊆ M is a kdimensional submanifold and Proof: The proof is inspired by [ Because Φ x is smooth we can differentiate it at g ∈ G and obtain a linear map T g Φ x : T g G → T Φx(g) M between vector spaces and a natural number k ∈ {0, . . . , dim M } such that We show that k is independent of the element g ∈ G. Let h ∈ G be another element. By the action property one has where we denoted the left multiplication with the element h on G by L h : G g → hg ∈ G. Since all the involved mappings are smooth, differentiating Eq. (2.1.11) gives by the chain rule (2.1.12) Since L h is a diffeomorphism it is also a local diffeomorphism, i.e. we have imT g L h = T hg G and since Φ h is a diffeomorphism we get by the same argument Thus if we set h = g g −1 we see that the rank of Φ x at g coincides with the rank of Φ x at any other point g ∈ G. Then the rank is indeed constant and the first claim is proved. As argumented before the Constant Rank Theorem implies then that the preimage Because the stabilizer is a closed subgroup, as the preimage of a closed subset {x} ⊆ M under a smooth function, it is also a Lie subgroup of G (c.f. [68,Theorem 20.10]). This proves the second claim. Now since Φ x is of constant rank, the third statement is just another consequence of the Constant Rank Theorem (c.f. [68,Theorem 7.13]). Then the next statement follows since there is a countable cover of G · x of k-dimensional submanifolds according to iii.). Finally, To prove the first property assume there is an element x ∈ (U · x) ∩ (C · x). Then, on one hand there is a g ∈ U such that Φ x (g) = x and on the other hand there is Thus h ∈ gG x ⊆ U G x which gives the contradiction. For the second property take x ∈ G · x and assume that That means C is closed and for this compact as a subset of a compact set G. We already mentioned that Φ x is continuous, so it is well known This construction is independent of the point x ∈ M . This concludes the proof.
Remark that G · x is not always a submanifold of M . One might think of the dense winding of the torus (c.f. [83, page 14]).
We want to stress that there is also a notion of right actions which works completely analog to left actions. One calls Φ : G × M → M a right action of G on M if for every x ∈ M and g, h ∈ G one has Φ(gh, x) = Φ(h, Φ(g, x)) and Φ(e, x) = x. We adapt the same notations and results we developed for left actions. It is intuitive to denote the orbit of an element x ∈ M in the case of a right action by x · G. In the following sections we need both kinds of actions but left actions more frequently. For this reason, we refer to left actions just as actions.
To conclude this section we mention the correspondence of Lie group actions and Lie algebra actions. A (left) Lie algebra action of a Lie algebra g on M is an anti-homomorphism of Lie algebras, while a right Lie algebra action is a homomorphism. Consider a Lie group action Φ : G × M → M and fix a x ∈ M . One can check that

Homogeneous Spaces
We start this section with the definition of a homogeneous spaces, an essential object of this thesis. After clarifying the smooth structure on homogeneous spaces we discover that there is always a natural transitive action on them and the corresponding stabilizers can be easily calculated. This can also be done by using an infinitesimal point of view   is a smooth surjective submersion.
Now it is easy to prove that the set G/H of orbits of R has the structure of a smooth manifold. One just has to check that R is free and proper.
Theorem 2.2.2 Let H be a closed subgroup of G. Then there is a unique way to structure the orbit space G/H as a smooth manifold such that the map is a smooth surjective submersion.
Proof: We already mentioned that the right multiplication R : H × G (h, g) → gh ∈ G is a right action of the Lie subgroup H on G and the corresponding orbits are the left cosets of H in G.
The action R is free, since for any h 1 , h 2 ∈ H and g ∈ G the equation gh 1 = R(h 1 , g) = R(h 2 , g) = gh 2 implies h 1 = h 2 . To prove that R is proper consider R : H × G (h, g) → (gh, g) ∈ G × G and an arbitrary compact subset K ⊆ G × G. If we also consider the usual right multiplication r : G × G (g 1 , g 2 ) → g 2 g 1 ∈ G on G and r : G × G (g 1 , g 2 ) → (g 2 g 1 , g 2 ) ∈ G × G one has Now r is a proper map since it is a diffeomorphism and for this r −1 (K) ⊆ G × G is compact. By assumption H ⊆ G is closed and so is is compact. This concludes the proof.
After discovering the smooth structure on G/H we introduce the following Let us discuss homogeneous spaces in detail. First we want to see that there is a natural action of G on G/H and that is action is transitive. The second step will be a kind of converse statement, namely that any transitive action on a manifold makes it into a homogeneous space. To achieve it, we need some preparation.
The statement that Eq. (2.2.2) defines a surjective submersion is very beneficial and we will use this in some proofs to get smoothness of maps. To do so we refer to the   Proof: First, Theorem 2.2.2 states that G/H is a manifold. Denote the map that is defined in Eq. (2.2.5) by Φ and let g 1 , g 2 ∈ G and g · H ∈ G/H be arbitrary. Then Φ(e, g · H) = (eg) · H = g · H and Φ(g 1 , Φ(g 2 , g · H)) = (g 1 (g 2 g)) · H = ((g 1 g 2 )g) · H = Φ(g 1 g 2 , g · H), i.e. Φ is a set theoretic action. Since the multiplication µ : G × G (g 1 , g 2 ) → g 1 g 2 ∈ G is smooth and pr defined in Eq. (2.2.2) is smooth according to Theorem 2.2.2, the map is smooth and the map id × pr : is a smooth surjective submersion. According to Theorem 2.2.4 the smoothness of Φ follows from the commutativity of the diagram for all g 1 , g 2 ∈ G. Thus Φ defined in Eq. (2.2.5) is indeed an action. It is also transitive since for arbitrary g 1 · H, g 2 · H ∈ G/H the element g 2 g −1 1 ∈ G satisfies Φ(g 2 g −1 1 , g 1 · H) = g 2 · H. If we take now an arbitrary element g · H, the corresponding stabilizer is given by We see that if this condition is satisfied there is an element h ∈ H such that gh = g g, which is equivalent to g = ghg −1 = Conj g (h) ∈ Conj g (H). Thus G g·H ⊆ Conj g (H). Conversely, any h ∈ H satisfies Φ(ghg −1 , g · H) = (ghg −1 g) · H = (gh) · H = g · H. (2.2.13) This implies G g·H = Conj g (H). Let us come to the infinitesimal part. Since pr : G → G/H is a submersion the map T e pr : T e G → T pr(e) (G/H) is surjective. By assumption T e G = g. We first show that h = ker T e pr. For any ξ ∈ h, one has exp(tξ) ∈ H for all t ∈ R and then pr(exp(tξ)) = pr(e) for all t ∈ R. Thus by the chain rule we have pr(e) = 0. (2.2.14) We showed h ⊆ ker T e pr. In fact, this is sufficient since pr : G → G/H is submersive and the dimension formula implies Then h = ker T e pr follows. By dividing by the kernel one gets the isomorphism defined in Eq. (2.2.7).
In the last theorem we have seen that there is a natural transitive action of G on any homogeneous space G/H. The next theorem gives the converse statement: if G acts transitively on M then M can be regarded as a homogeneous space G/H where H is the stabilizer of an arbitrary point of M .
is a diffeomorphism, which commutes with the action of G on G/G x . Moreover, all stabilizers are isomorphic via Proof: Parts of the proof and the notation are inspired by [83,Theorem I.3.3]. Let x ∈ M be an arbitrary point. We already mentioned that G x ⊆ G is a closed subset. Then G/G x is a homogeneous space according to Theorem 2.2.2. If we take any g ∈ G and h ∈ G x we know on the one hand that β x (h · G x ) = Φ(h, x) = x and on the other hand that since the projection is a smooth surjective submersion and Φ x : G → M is smooth. The next step is to show that β x is a bijective immersion. Then it has to be a diffeomorphism since we have already proven its smoothness (c.f. [68,Corollary 7.11]). Thus consider Denote the Lie algebra corresponding to G x by g x (remember that G x is a Lie group according to Theorem 2.1.1 ii.)) and take ξ ∈ g. We can identify a equivalence class [ξ] ∈ g/g x with T e pr[ξ] ∈ T pr(e) G/G x according to Proposition 2.2.5. Thus if we assume [ξ] ∈ ker T pr(e) β x , one has exp(tξ) ∈ G x for any t ∈ R, since exp(0ξ) = e ∈ G x and Thus ξ ∈ g x for which [ξ] = 0 and the map defined in (2.2.20) is injective. For g ∈ G, the derivative of the equality gives according to the chain rule, where g −1 : Since Φ g and g −1 are diffeomorphisms this implies the injectivity of T pr(g) β x . As we argued this is enough to show that β x is a diffeomorphism since the involved spaces are second countable. Moreover, β x is equivariant, i.e. it commutes with Φ, since for any g ∈ G and h · G x ∈ G/G x one has whereΦ denotes the action of G on G/G x defined by Eq.
To prove that all stabilizers are isomorphic via the map defined in Eq. (2.2.16) we choose two points which means that Conj g (g ) ∈ G Φg(x) . Thus the map defined in Eq. (2.2.16) is well-defined. Conj g is obviously smooth and also invertible with inverse Conj g −1 :  where 1 ∈ SO(3) denotes the identity matrix. This action is even transitive: choose two points p = q on S 2 . Then there is a unique plane E in R 3 such that 0, p, q ∈ E, where 0 = (0 0 0) T ∈ R 3 . Choose a vector 0 = n ∈ R 3 perpendicular to E. There is an angle α ∈ [0, 2π] such that a rotation of E around n transfers p onto q. After a change of coordinates we can assume that this is a rotation around the z-axis and p lies on the x-axis. Then satisfies Ap = q. If p = q ∈ S 2 one simply has 1p = q. Altogether Φ is transitive. Since any 1 = A ∈ SO(3) has an eigenvalue 1 = λ ∈ C, only 1 induces the identity diffeomorphism on S 2 and Φ is effective in addition. We want to compute the stabilizer at the point e 1 = (1 0 0) T ∈ S 2 .
Thus we are searching for the elements A = If we restrict Φ to SO(3) × S 2 the action reads Φ(A, p) = Ap ||Ap|| = Ap, since ||Ap|| = 1 as it was shown in the last example. Thus Φ is indeed an extension of the action in Example 2.2.7.
This shows immediately the transitivity of Φ. It is also effective, since any 1 = A ∈ SL(3, R) has an eigenvector 1 = λ ∈ C. Consequently, S 2 ∼ = SL(3, R)/SL(3, R) p for any p ∈ S 2 , where the stabilizer SL(3, R) p has to be 6-dimensional since dim(SL(3, R)) = 8. Take for example e 1 = (1 0 0) T ∈ S 2 . Then the corresponding stabilizer is To prove this, consider an arbitrary matrix B =  For the next example we remind the reader of the notion of Minkowski space which is common in special relativity. Consider the matrix x k y k ∈ R (2.2.33) associated to the matrix (2.2.32) is said to be the Minkowski space. The corresponding quadratic form is The set O(1, 3, R) of isometries of the Lorentz metric ϕ 1,3 is said to be the Lorentz group. Thus that fulfil det(Λ) = 1 is further denoted by SO(1, 3, R). One often writes x = x 0 x ∈ R 4 to indicate the time part x 0 ∈ R and the space part x ∈ R 3 of the vector x.  where pr is the projection on the space part of a vector, i.e. pr(( The map Φ is obviously well-defined and smooth. For any x ∈ S 2 one has since || x|| = 1. Proving the action property and effectiveness of Φ is a bit more involved. We want to give another argument to prove that there is a transitive action of L ↑,+  where To get a maximal abelian subalgebra a of p one calculates the commutator of A = 0 a T a 0 3×3 Thus, it is easy to see that is a maximal abelian subalgebra of p. A nilpotent Lie subalgebra n of so(1, 3) given as the sum of root spaces of a choice of positive roots on a is given by If we denote the corresponding connected Lie groups by K, A and N we get the Iwasawa decomposition of L ↑,+ 3 , namely Since [a, n] ⊆ n we know that a ⊕ n is a Lie subalgebra of so (1,3). Thus AN is a subgroup of L ↑,+ 3 and we showed that S 2 can be structured as a homogeneous space via a transitive action We see in Section 2.4 that Example 2.2.7, Example 2.2.8 and Example 2.2.9 provide the only three (non-equivalent) ways to structure S 2 as a homogeneous space via an effective action.

The Euler Characteristic of Compact Homogeneous Spaces
In Proposition 2.2.5 and Theorem 2.2.6 we gave a characterization of homogeneous spaces by quotient spaces of Lie groups that act transitively on the homogeneous space modulo a stabilizer of the action. There is another way to detect homogeneous spaces with a topological background.
For k ∈ N 0 we denote the set of smooth k-differential forms on a smooth manifold M by is said to be the de Rham cohomology of M .  This is our tool to detect homogeneous spaces. A homogeneous space M is compact if for example the Lie group G that acts transitively on M is compact. We prove that in this situation the Euler characteristic of M is non-negative, following [93,Section 5]. We need to sum up some basic definitions and properties of Lie group representations first.
where GL(V ) denotes the general linear group of V consisting of the bijective linear maps V → V with concatenation as group multiplication.
The map Ad * : G → GL(g * ) that assigns an element g ∈ G the adjoint of the linear map Ad(g −1 ) is said to be the coadjoint representation of G on g * . For any k ∈ N one can extend those two maps linearly to obtain representations Λ k Ad and Λ k Ad * of G on Λ k g and Λ k g * , respectively. Now choose an arbitrary representation (π, V ) of G. If there is a scalar product ·, · on V such that for all g ∈ G and v, w ∈ V the representation (π, V ) is said to be orthogonal. A real vector space can always be endowed with a scalar product. We want to prove that there is a scalar product on V satisfying (2.3.5) for a chosen representation (π, V ) if G is compact. The first step is to consider a special measure. Take a n-dimensional manifold M and a volume form ω on it, i.e.
The measure µ corresponding to ω is then defined by the relation for all f ∈ C ∞ (U ). One can check that this is independent of the chart (U, x i ). In the case of a Lie group G we choose a non-zero elementω ∈ Λ n (T e G) and define ω = T e L gω for any g ∈ G. If G is compact in addition we can normalize ω such that the corresponding measure fulfils It has been proved in [45] that this measure is unique and bi-invariant, i.e.
for all f ∈ C ∞ (G) and h ∈ G. We call it the normalized Haar measure on G. Now we come back to orthogonal representations: Proposition 2.3.5 Let (π, V ) be a representation of a compact Lie group G and ·, · any scalar product on V . Then is a scalar product on V such that Eq. (2.3.5) holds, i.e. (π, V ) is a orthogonal representation of G.
Proof: Clearly Eq. (2.3.9) defines a scalar product on G. It fulfils (2.3.5) since for g , h 1 , h 2 ∈ G one has Also remark the notion of maximal tori: Definition 2.3.6 Let G be a Lie group. Then a Lie subgroup T is said to be a i.) torus in G if T is isomorphic to a product S 1 × · · · × S 1 of 1-spheres.
ii.) maximal torus in G if for any torus T in G such that T ⊆ T one has T = T .
These objects are of fundamental use in the theory of connected compact Lie groups. Some properties are collected in the following Proposition 2.3.7 Let G be a connected compact Lie group.
i.) Any tours in G is contained in a maximal torus in G.
ii.) Any element in G is contained in a maximal torus in G.
iii.) Any two maximal tori in G are conjugated to each other, i.e. if T and T are maximal tori in G there is a g ∈ G such that For this one calls the dimension of any maximal torus in G the rank of G.
iv.) Any maximal torus T in G has a generating element g 0 ∈ T , i.e. for any g ∈ T there is a natural number k ∈ N 0 such that g = g k 0 .
All statements are taken from [ Furthermore, one has a topological result on representations of compact Lie groups. In analogy to Definition 2.3.16 we want to define the Euler characteristic of a representation. Let (π, V ) be a representation of a compact Lie group G with n = dim(V ). Then for any 1 ≤ k ≤ n the map π induces a representation Λ k π on Λ k V . Denote by I k the subspaces of Λ k V consisting of the elements v that satisfy (Λ k π(g))(v) = v for all g ∈ G. Then define the Poincaré polynomial P π of (π, V ) by where λ k = dim(I k ) are the Betti numbers corresponding to π. We know from representation theory (c.f. [28, Section 12.6.2]) that the Betti numbers are obtained by integration of the trace of the corresponding representation over the whole group, i.e.
be a representation of a compact Lie group G. The Euler characteristic χ(π) of (π, V ) is defined by In the end we want to prove that the Euler characteristic of a connected compact homogeneous space is non-negative. The next lemma is a step towards this: Lemma 2.3.9 Let (π, V ) be a representation of a connected compact Lie group G. Then χ(π) ≥ 0. Moreover, one has χ(π) > 0 if and only if for each maximal torus T in G one has π(g)v = v for all g ∈ G, only for a v ∈ T if v = 0.
Proof: The discussion above implies resulting from a principal minor argument. According to Proposition 2.3.5 we can assume that π(g) is a orthogonal transformation for all g ∈ G. Since G is connected π(g) is also orientation preserving. First assume that n is odd. Then according to [43,Lemma 7.3.1] π(g) has an eigenvalue λ = 1, i.e. det(−π(g) + id V ) = 0. Thus in this case χ(π) = 0. If n is even one has the normal form of orthogonal matrices for some φ i ∈ R. If at least one −1 appears in the matrix (2.3.15) one has det(−π(g) + id V ) = 0. Since det(R i ) = 1 and π(g) preserves orientation there is a even number of 1 appearing in (2.3.15) and since we have Thus χ(π) ≥ 0 and we have proven the first claim of the proposition. We prove both directions of the second statement by contraposition. Assume Choose any maximal torus T in G with generating element g 0 ∈ G. Then of course det(−π(g 0 )+id V ) = 0 which means that 1 is an eigenvalue of π(g 0 ), i.e. π(g 0 ) fixes a non-zero vector v ∈ V . Since π is a Lie group homomorphism and any other element g in T is obtained as a power p ∈ N of g 0 one has π(g)v = π(g p 0 )v = π(g 0 ) · · · π(g 0 )v = v and every element of T fixes v. Conversely, assume that there is a maximal torus T fixing a non-zero vector v ∈ V , i.e. det(−π(g) + id V ) = 0 for all g ∈ T . Now any element g ∈ G is conjugated to an element g 0 ∈ T , i.e. there is an h ∈ G such that g = hg 0 h −1 . Via cyclic permutations of the arguments of the determinant this gives together with the homomorphism property of π This proves det(−π(g) + id V ) = 0 for all g ∈ G and χ(π) = 0.
The next proof deals with differential forms on a manifold that are invariant under the action of a Lie group that acts on the manifold. We denote by α a differential k-form on a smooth manifold M and we say that it is invariant under the action Φ of a Lie group G on M if i.e. for all p ∈ M , g ∈ G and all tangent vectors X 1 , . . . , X k ∈ T p M one has We use the notation to denote set of k-forms on M that are invariant under the action Φ. The corresponding k-th invariant de Rham cohomology is then defined by Proof: Choose an element g ∈ G. Since G is connected there is a smooth curve γ : [0, 1] → G in G such that γ(0) = e and γ(1) = g. Furthermore one defines the smooth map coincide on cohomology level. This construction is valid for all g ∈ G. In a next step we consider the map for any α ∈ Ω • (M ), by the bi-invariance of the normalized Haar measure. We want to show that which implies thatΞ is surjective. By Eq. (2.3.23) one has for any i.e.Ξ is injective. This concludes the proof.

Now consider a connected homogeneous space
We denote the elements that are invariant under this action by Then one can prove the following Lemma 2.3.11 Let M = G/H be a connected homogeneous space such that G is connected and compact, where we denote the transitive action of G on M by Φ. Then Proof: Consider the map It is easy to see that F (α e ) ∈ Ω • (M ) G,Φ for any α e ∈ (Λ • T * e M ) H . To prove that this map is welldefined one has to argue that Φ g (e ) = Φ h (e ), for any g, h ∈ G and Since Φ is transitive it is also locally transitive, i.e. the corresponding Lie algebra action φ| p : g → T p M is surjective for all p ∈ M . Thus, for any v p ∈ T p M there is a ξ ∈ g such that the fundamental vector field ξ M for ξ on M satisfies for any p ∈ M . Applying this to every tensor component of an arbitrary X p ∈ Λ k T p M , one obtains an element ξ = ξ 1 ∧ · · · ∧ ξ k ∈ Λ k g such that We denote the left hand side of Eq. (2.3.31) also by ξ M . Let α e ∈ (Λ k T * e M ) H , X e ∈ Λ k T e M and h ∈ H be arbitrary. Then where we used fundamental equations of the Cartan calculus. This shows . Since e = Φ e (e ) this implies To prove that F is surjective choose an arbitrary and since T e Φ g is a linear isomorphism this implies for any p ∈ M that where g ∈ G such that Φ g (e ) = p. This concludes the proof.
Now we can prove that the Euler characteristic of a connected compact homogeneous space is non-negative if there is a connected compact Lie group acting transitively on it.
where C k (M ) G,Φ and B k (M ) G,Φ denote the set of all invariant k-cocycles and k-coboundaries on M , respectively. Then it follows from Lemma 2.3.10 that where we used Eq. (2.3.32) in ( * ). But since there is the representation Λ • Ad * of the connected compact Lie group H on Λ • T * e M Lemma 2.3.9 implies Also the statement about the maximal torus can be deduced from Lemma 2.3.9.
The next step consists in extending this result to connected non-compact Lie groups. This is in general very hard to obtain. The situation gets easier if one assumes the space M to be simply connected in addition, what will be enough for considering the sphere S 2 as an example.
Nevertheless, we cite the proofs for the case when M is only assumed to be connected since we are also interested in the pretzel surfaces that are not simply connected. We want to get a tool to use Proposition 2.3.12 also in the case of non-compact Lie groups G if M is simply connected. If G is connected and M simply connected, any stabilizer G x , where x ∈ M , of the transitive action of G on M has to be connected since M = G/G x . This means we can use the following famous theorem of Cartan for each, G and G x (c.f. [56, Theorem 2]): Theorem 2.3.13 Every connected Lie group G is a Cartesian product of a maximal compact Lie subgroup K ⊆ G and an Euclidean vector space E, i.e.
Assume that M = G/G x is simply connected. Then there are compact Lie subgroups K, L and Euclidean vector spaces E, F such that G = K × E and G x = L × F according to Theorem 2.3.13. Since G x ⊆ G and any two maximal compact subsets of G are conjugate to each other we can assume L ⊆ K. Now G/L can be seen as the total space of a vector bundle once with basis G/K and fibre K/L and twice with basis G/G x and fibre G x /L. For example π : G/L → G/K maps each coset of L in G to the coset of K in G which contains it. But since G/K ∼ = E is an Euclidean space, a general topological result says that this vector bundle is trivial, i.e.
The same holds for the second vector bundle, which implies that there is a global section There is an even more general result due to A. L. Onishchik and V. V. Gorbatsevich which shows that the assumption of M being simply connected is not necessary (c.f. [83, Part II, Chapter 5, Theorem 1.2]).

The Classification of Transitive Actions on S 2
The goal of this section is to describe all ways one can structure the 2-sphere as a homogeneous space. As we discovered in Section 2.2 it is equivalent to determine all transitive actions on S 2 .
This classification has to be done with a notion of equivalent actions.
for all g ∈ G and x ∈ M . They are called locally equivalent if there is an open subset U ⊆ M such that Φ| G×U and Φ | G×U are equivalent.
It is clear by definition that equivalent actions are locally equivalent. It is also quite intuitive that a classification makes only sense if one forces the actions to be effective. Otherwise one could enlarge the Lie group by adding elements that simply induce the identity diffeomorphism on the manifold. It is reasonable to classify transitive Lie groups such that every element induces a different diffeomorphism, i.e. actions that are effective in addition. In Chapter 6 we realize that this assumption is no restriction at all for our purpose. In the 1960's many classifications of transitive effective actions appeared. Pioneer work was done by A.L. Onishchik. He considered compact homogeneous spaces and gave classifications of compact Lie groups that act as well as for non-compact Lie groups (see e.g. [80][81][82][83]). In [83, Theorem 2.6] he classified in particular all locally effective transitive actions of connected compact Lie groups on simply connected homogeneous spaces of rank 1. We follow a source Onishchik also uses, namely the paper [74] of D. Montgomery and H. Samelson. There they classify all connected compact Lie groups that act transitively and effectively on S n , n ∈ N. We want to mention that the classification splits into even and odd n. But since we are just interested in n = 2 we do not see these effects. We first prove that if G acts in the way described above it already has to be simple. Then there is just a list of Lie groups left to check. For this we need a little Lemma 2.4.2 Let G 1 , G 2 be connected compact Lie groups and N a finite normal subgroup of acts transitively on S 2 then for one i = 1, 2 and x ∈ S 2 the Lie group G i /(G i ∩ G x ) acts transitively on S 2 .
Proof: The proof and the notation is taken from [74,Theorem I']. If G acts transitively on S 2 then of course also G acts transitively on it. Take the stabilizer G x of this action of any point x ∈ S 2 . It is connected because S 2 is simply connected and compact. Moreover, any element h ∈ G x can be written as h = h 1 × h 2 with some h 1 ∈ G 1 and h 2 ∈ G 2 and is contained in a maximal toral subgroup T ⊆ G x . It is proven in [92] that G and G x have the same rank, which implies that T ⊆ G, thus T = T 1 × T 2 , where T 1 and T 2 are maximal toral subgroups of G 1 and G 2 , respectively. This implies that and Proof: Assume that the connected compact Lie group G acts transitively and effectively on S 2 but is not simple. Since G is a connected and compact Lie group there are two connected compact subgroups G 1 and G 2 of G and a finite normal Again we just want to sketch the proof: Onishchik proves this theorem for all compact homogeneous spaces with positive Euler characteristic and also gives a decomposition of the Lie group into simple normal divisors (c.f. [82,Theorem 2]). The proof bases on the classification of the t-subalgebras and k-subalgebras of all semisimple Lie groups which was also done by Onishchik in [81]. In this paper he constructs standard tand k-subgroups and shows that any tand k-subgroup of a semisimple Lie group is conjugate to a standard one on Lie algebra level (c.f. [81, Theorem 3 and Theorem 4]).
Since semisimple Lie groups are well known and classified one can conclude the next theorem (c.f. [82,Theorem 3]).
Theorem 2.4.6 Any connected Lie group that acts transitively and effectively on S 2 is equivalent to SO(3), SL(3, R) or L ↑,+

.
Remark that the actions mentioned in Theorem 2.4.6 are given in Example 2.2.7, Example 2.2.8 and Example 2.2.9, respectively. This is also consistent with Theorem 2.3.14 since SO (3) is a compact subgroup of L ↑,+ 3 and SL(3, R) that acts transitively on S 2 .

Lie Bialgebras and r-Matrices
We discussed the geometry of homogeneous spaces and their connection to transitive Lie group actions. In a way, it is natural to study those Lie groups in order to learn more about homogeneous spaces. In Subsection 2.4 we showed that for some examples only very few Lie groups are relevant. In general, if we ask the homogeneous space to have additional structures the situation is of particular interest and it is natural to ask whether there is an impact on the corresponding Lie group. The converse might be even more interesting: is there a structure on a Lie group that acts on a manifold, that forces the manifold to be a homogeneous space? It turns out that we have to treat this on infinitesimal level, i.e. on the corresponding Lie algebra. These ideas are made concrete in Chapter 5, but shall suffice as a rough motivation right now.
The structures we mentioned are r-matrices. Those objects where developed in the course of integrable lattice systems. This is because r-matrix theory gives rise to Lax equations (see e.g. [18]). In order to motivate the notion of r-matrices we embed them in the theory of Lie bialgebras, which are themselves interesting for several reasons: first they occur as infinitesimal objects of Poisson-Lie groups that are involved again in integrable systems via monodromy matrices (consider for example [7]). We state the connection between Lie bialgebras and Poisson-Lie groups in Section 3.2 without proving this statement due to V. G. Drinfel'd. Second, Lie bialgebras are semi-classical limits of quantum groups and conversely any Lie bialgebra can be quantized in that way. This is the famous theorem by P. Etingof and D. Kazhdan (c.f. [38]). In the same manner R-matrices appear as quantizations of r-matrices (c.f [39, Theorem 9.2]) and the quantum Yang-Baxter equation as a quantization of the classical Yang-Baxter equation (c.f. [39,Theorem 9.3]). We introduce the classical Yang-Baxter equation as a measure for rmatrices.
For Lie bialgebras we need some general Lie algebra cohomology and in particular the Chevalley-Eilenberg complex which we develop in Section 3.1 together with some well-known results in this topic. Then we are able to make a motivated definition and classification of Lie bialgebras in Section 3.2. Some interesting examples of Lie bialgebras finally lead to r-matrices in Section 3.3. We classify them and also consider the case when the corresponding Lie algebra is simple what will be helpful in the later sections. As a tool we define the Etingof-Schiffmann subalgebra in Section 3.4, a finite-dimensional Lie subalgebra in which the r-matrix is non-degenerate.

Lie Algebra Representations and Cohomology
Working with Lie algebras is interesting because they can act on other structures. We say that a Lie algebra (g, [·, ·]) over a field k acts on a k-vector space V if for every x ∈ g there is a linear map ρ x : V → V such that the assignment ρ : g x → ρ x ∈ gl(V ) is a Lie algebra homomorphism, i.e. for all x, y ∈ g one has where we endowed the general linear Lie algebra gl(V ) with the commutator. In this case we say that (ρ, V ) is a representation of g or if we want to emphasize that the Lie algebra g acts on V we call V a g-module. We also use the short notation The following is inspired by [103]. i.) First of all, g can act trivially on k, i.e. x.λ = 0 ∈ k for all x ∈ g, λ ∈ k. In this way k becomes a g-module, called the trivial module.
ii.) Furthermore, g can act on itself by the adjoint action, i.e. ad x (y) = [x, y] for x, y ∈ g. Indeed, g becomes a g-module in this way because the Lie bracket is bilinear by definition and Eq. (3.1.1) is just the Jacobi identity in this case.
If not stated differently we always view g ⊗ · · · ⊗ g as this g-module (see Chapter 3.3). For example for n = 2 we get This motivates the notations Having in mind these examples we consider again some arbitrary g-module V , where g is a Lie algebra over a field k of characteristic zero. For a natural number n ∈ N we denote by C n (g, V ) = Hom k (Λ n g, V ) the vector space of k-linear maps from the n-th exterior power The elements of C n (g, V ) are called n-cochains on g with values in V (or just n-cochains if there is no confusion which Lie algebra module is meant). We define a mapping δ n : C n (g, V ) → C n+1 (g, V ) from the set of n-cochains to the set of (n + 1)-cochains by for all c ∈ C n (g, V ) and x 1 ∧ · · · ∧ x n+1 ∈ Λ n g. The point in the first sum stands of course for the representation ρ corresponding to the g-module structure of V , i.e. x k .c(x 1 ∧· · ·∧x k ∧· · ·∧x n+1 ) = ρ x k (c(x 1 ∧ · · · ∧x k ∧ · · · ∧ x n+1 )) andx k denotes that the element x k is removed in the exterior product. In particular, we are interested in the kernel of δ n . We denote this vector space by Z n (g, V ) = ker(δ n ) and call its elements n-cocycles on g with values in V or just n-cocycles. This space can be compared to the image of δ n−1 , i.e. the vector space B n (g, V ) = im(δ n−1 ) whose elements are said to be the n-coboundaries on g with values in V or again just n-coboundaries, where we set B 0 (g, V ) = {0}. The reader can check that δ n+1 • δ n = 0. We often just write δ = δ n . The equation , i.e. every n-coboundary is a n-cocycle. Then the set is a well-defined k-vector space.
Definition 3.1.2 Let (ρ, V ) be a Lie algebra representation of g. The tuple (C n (g, V ), δ n ) n∈N 0 is said to be the Chevalley-Eilenberg complex and δ n = δ the Chevalley-Eilenberg differential. Moreover, one calls H n (g, V ) the n-th cohomology group on g with values in V , or just the n-th cohomology group.
for all x ∈ g. This vector space is sometimes denoted by V g . Thus the 0-th cohomology group consists of the elements of V that are annihilated by all elements of g via the corresponding g-module representation, i.e.
If g acts trivially on k we obtain H 0 (g, k) = k. If g acts on g by the adjoint action we get and if g acts on g ⊗ g by ad (2) we see that where the quotient of derivations and inner derivations results in the so-called outer derivations Out(g, V ) on g with values in V . In particular, if g acts trivially on k we obtain where the star denotes the dual space and [g, g] the k-span of all brackets [x, y], for some x, y ∈ g.
iii.) For n = 2 one can prove (c.f [78, Proposition I. 3.]) that where Ext(g, V ) is the set of equivalence classes of abelian extensions of g by V . If g is a simple Lie algebra over C and Ω ∈ g⊗g the Casimir element associated to a non-degenerate invariant form on g we get (Λ 3 g) g = CZ, Many details about Lie algebra cohomology can be found in [51], [78] and [84]. To end this section we want to state some famous results about Lie algebra cohomology.
Theorem 3.1.4 Let g be a complex semisimple Lie algebra and V a finite-dimensional g-module. Then Proof: The first statement is known as Whitehead's Lemma and can be found in [102]. The second statement is a consequence of this lemma (c.f. [52]).
Remark 3.1.5 In [24] it is proven that if g is a nilpotent Lie algebra of dimension n ∈ N the dimension of H k (g, k) is at least 2 for 1 ≤ k ≤ n − 1. This gives a helpful intuition that nilpotent Lie algebras have "big" Lie algebra cohomologies while the Lie algebra cohomologies of semisimple Lie algebras are "small". Finally, for a semisimple Lie algebras there is a splitting of H n (g, V ) into 1-dimensional modules with respect to a root decomposition of g. This is discussed in [64].

Lie Bialgebras and Poisson-Lie Groups
Lie bialgebras and Poisson-Lie groups are well-studied objects. We refer to the first section of the notes [62]  Consider a Lie algebra g over a field k of characteristic zero. We want to equip g with some additional structure. As motivated from the last section we regard a linear map that is a 1-cocycle on g with values in g ⊗ g, where we view g ⊗ g as a g-module via ad (2) as usual, i.e. γ ∈ Z 1 (g, g ⊗ g). It is common to use the short notation for x ∈ g to represent the element γ(x) ∈ g ⊗ g. The cocycle condition reads or equivalently γ([x, y]) = ad (2) x (γ(y)) − ad (2) y (γ(x)), (3.2.4) for all x, y ∈ g. If we require γ to satisfy in addition and Alt((γ ⊗ 1)γ(x)) = 0 (3.2.6) for all x ∈ g, where Alt is the k-linear function 3 g → 3 g that satisfies on factorizing tensors Alt(x ⊗ y ⊗ z) = x ⊗ y ⊗ z + y ⊗ z ⊗ x + z ⊗ x ⊗ y, the Lie algebra g becomes also a Lie coalgebra, or equivalently the dual g * of g gets a Lie algebra. The Lie bracket on g * is given by the left transpose of γ, i.e. is a Lie bracket on g * . Indeed we can define a bracket [·, ·] g * : g * ⊗ g * → g * on g * by for all ξ, η ∈ g * . If we introduce the non-degenerate dual pairing ·, · : n g * × n g → k we can prove that Eq. (3.2.8) defines a Lie bracket on g * if and only if γ satisfies conditions (3.2.5) and (3.2.6). Of course condition (3.2.5) can immediately be absorbed if we require γ to be a map g → g ∧ g. The skew-symmetry of [·, ·] g * follows from condition (3.2.5) because for all x ∈ g, ξ, η ∈ g * one has This implies [ξ, η] g * = −[η, ξ] g * because the dual pairing is non-degenerate. Arranging the above equations in another order we see that the skew-symmetry of [·, ·] g * implies condition (3.2.5). Moreover, the Jacobi identity of [·, ·] g * is equivalent to condition (3.2.6), the so called coJacobi identity. To prove this, observe that for x ∈ g and ξ, η, ζ ∈ g * one has Thus every term of the coJacobi identity corresponds to one term of the Jacobi identity of [·, ·] g * and the equivalence is proved. After recognizing the powerful impact of the above considerations we give the well-motivated definition of a Lie bialgebra. satisfying the coJacobi identity (3.2.6) and the cocycle identity (3.2.4). The cocycle γ is called the Lie bialgebra structure of (g, [·, ·], γ).
We already proved the following proposition in the above lines.
Every new object goes hand in hand with its morphisms and substructures.
i.) A Lie algebra morphism φ : g → k is said to be a morphism of Lie bialgebras if it respects the cocycles, i.e.

2.11)
for all x ∈ g. The condition being a Lie algebra morphism reads

2.12)
for all x, y ∈ g.
ii.) A Lie subalgebra h of g is said to be a Lie subbialgebra if the cocycle on g restricts to a map iii.) A Lie ideal h of g is said to be Lie coideal if 14) The condition being a Lie ideal of g is that h is a left ideal of g with respect to the Lie bracket (or equivalently a right ideal or an ideal), i.e. [h, g] g ⊆ h.
Remark that if h is a Lie ideal of (g, [·, ·]) the quotient Lie algebra (g/h, [·, ·]) inherits a Lie bialgebra structure from g if and only if h is a Lie coideal. We already realized in Proposition 3.2.2 that the cocycle γ : g → g ∧ g on g that satisfies the coJacobi identity is equivalent to a Lie bracket on the dual g * of g. We expand this result in the sense that if γ satisfies the above conditions and the cocylce condition, not only g but also g * has the structure of a Lie bialgebra. Thus the notion of Lie bialgebra is totally self dual.
Proof: We already proved that (g * , t γ) is a Lie algebra if and only if γ satisfies conditions (3.2.5) and (3.2.6). The only thing left to prove the proposition is that t [·, ·] is a cocycle. The restriction to finite-dimensional Lie bialgebras is necessary if we want t ( t [·, ·]) to define the Lie bracket [·, ·] on (g * ) * = g again. Let us compute for x, y ∈ g and ξ, η ∈ g * = ξ ∧ η, (ad (2) x (γ(y)) − ad (2) y (γ(x))) = ξ ∧ η, ( x, y [1] ∧ y [2] + y [1] ∧ x, y [2] − y, x [1] The claim follows because the dual pairing is non-degenerate. Then t [·, ·] is a cocycle on g * if γ is a cocylce on g. Using the equations above in different order one can check that if t [·, ·] is a cocycle on g * also γ is a cocylce on g and in the finite-dimensional case these are indeed equivalent conditions. A nice pictorial proof of this statement can be found in [ By the bilinearity and antisymmetry of [·, ·] condition (3.2.16) determines the Lie bracket completely. Define a map γ : g → g ∧ g by γ(X) = 0, γ(Y ) = −X ∧ Y and linear expansion and prove that this structures (g, [·, ·]) as a Lie bialgebra. Of course it is enough to prove this just for the basis elements. Indeed, the cocycle condition holds: We also check the coJacobi identity for X Thus (g, [·, ·], γ) is a Lie bialgebra. To answer this we give a very short repetition on Poisson manifolds. For a conceptual introduction consider for example [67], [101] or [105]. A smooth manifold M together with a Poisson bracket class of examples of Poisson manifolds is given by symplectic manifolds. Remember that a symplectic manifold is a smooth manifold M together with a closed non-degenerated 2-form ω ∈ Γ ∞ (Λ 2 T * M ), called the symplectic structure. In fact, for any symplectic structure ω ∈ Γ ∞ (Λ 2 T * M ) we can always define a Poisson bracket by The objects X f and X g in Eq. (3.2.23) are said to be the Hamiltonian vector fields of f and g, respectively and they are defined by

2.24)
where : T * M → T M is the musical isomorphism. Thus another way of writing Eq. (3.2.23) is

2.25)
for f, g ∈ C ∞ (M ). It is known that for any p ∈ M the map is surjective if (M, ω) is a symplectic manifold and that holds for all f, g ∈ C ∞ (M ). An interesting structure occurs if we combine the concept of Poisson manifolds with the notion of Lie groups. Remark 3.2.7 Let (G, {·, ·}) be a Poisson-Lie group and denote by λ u : G v → uv ∈ G and ρ u : G v → vu ∈ G the left and right multiplication on G with some element u ∈ G, respectively. Then the group multiplication m is a Poisson map, i.e.
where f, g ∈ C ∞ (G) and u, v ∈ G. There is also an equivalent condition to (3.2.28) for the corresponding Poisson bivector π ∈ Γ ∞ (Λ 2 T G), namely

2.29)
for all u, v ∈ G. In particular, we see that if G is a Poisson-Lie group with identity element e ∈ G we get π(e) = 0.  between the categories of simply connected Poisson-Lie groups and finite-dimensional Lie bialgebras and this functor gives an equivalence of these categories.
Thus, one can see Lie bialgebras in a bigger context. The proof is not necessary for our purpose and we just refer to [39,Theorem 2.2]. For instance, this theorem motivates the notion of the dual G * of a Poisson-Lie group G , where g * = Drin(G * ) is the dual of g = Drin(G). In the next section we see some special kinds of Lie bialgebras. They have global counterparts in conformity with the functor of the last theorem. Consider [39, Section 9.4] for the definitions of these so called coboundary Hopf algebras and the connection to their infinitesimal objects.

Coboundary Lie Bialgebras and the Classical Yang-Baxter Equation
Here we give a nice connection of the last two sections. , γ) with a cocycle γ on g with values in g ⊗ g. How does the situation change when γ is also a coboundary on g with values in g ⊗ g, i.e. if there is an element r ∈ C 0 (g, g ⊗ g) = g ⊗ g such that γ = δr?
And what conditions has an arbitrary element r ∈ g ⊗ g to obey to define a cocycle γ = δr such that (g, [·, ·], γ) becomes a Lie bialgebra? Because δ 2 = 0 the element δr will automatically be a cocycle, but conditions (3.2.5) and (3.2.6) do not have to be valid in the general case. A powerful instrument to measure if the coJacobi identity is also given is the so called classical Yang-Baxter map CYB : g ⊗ g → g ⊗ g ⊗ g = g ⊗3 defined by for all elements r = r 1 ⊗ r 2 = r 1 ⊗ r 2 ∈ g ⊗ g (we use a notation very similar to (3.2.2)), where we defined r 12 = r 1 ⊗ r 2 ⊗ 1, on the universal enveloping algebra U(g) of g. The brackets in (3.3.1) are intuitive abbreviations defined by [r 12 , r 13 ] = r 1 , r 1 ⊗ r 2 ⊗ r 2 , where we denoted the second summation by r = r 1 ⊗ r 2 . With this map we are able to determine whenever r gives rise to a Lie bialgebra via the above construction. Theorem 3.3.1 Given a Lie algebra (g, [·, ·]) and an element r ∈ g ⊗ g that is skew-symmetric. The triple (g, [·, ·], δr) is a Lie bialgbra if and only if CYB(r) ∈ g ⊗3 is a g-invariant element. The condition for CYB(r) to be g-invariant reads ad (3) x (CYB(r)) = [x, CYB(r)] = 0,

3.2)
for all x ∈ g.
To separate the two summations over the tensor components of r we denoted the second one by r 1 ⊗ r 2 . After applying Alt we obtain twelve terms. The skew-symmetry of [·, ·] and r imply Alt((γ ⊗ 1)γ(x)) = [x, r 1 ], r 1 ⊗ r 2 ⊗ r 2 − r 1 ⊗ [x, r 2 ], r 1 ⊗ r 2 + r 1 ⊗ r 1 ⊗ [x, r 2 ], r 2 We know that CYB(r) = r 1 , r 1 ⊗ r 2 ⊗ r 2 + r 1 ⊗ r 2 , r 1 ⊗ r 2 + r 1 ⊗ r 1 ⊗ r 2 , r 2 , so the right hand side of Eq. (3.3.3) reads ad (3) x (CYB(r)) = x, r 1 , r 1 ⊗ r 2 ⊗ r 2 + r 1 , We apply the Jacobi identity to the terms with double Lie bracket and get These are exactly the twelve terms that we obtained on the left side times minus one. We only have to interchange the summation over the tensor components of r in some terms, i.e. we have to switch r 1 ⊗ r 2 and r 1 ⊗ r 2 , which is allowed since these summations are all finite. Formula  x (CYB(r)) = 0 for all x ∈ g. This concludes the proof.
Following the spirit of this thesis we define the corresponding morphisms and substructures.
In this case r is said to be a r-matrix of g. Condition ii.) is called the classical Yang-Baxter equation.
It follows from the definition above that triangular Lie bialgebras, and as a consequence also r-matrices, are in one-to-one correspondence to solutions of the classical Yang-Baxter equation in Λ 2 g. Furthermore, every triangular structure is quasitriangular and every triangular Lie bialgebra has also the structure of a quasitriangular Lie bialgebra. As a first example of triangular Lie bialgebras and r-matrices we go back to Example 3.2.5. There we endowed any two-dimensional noncommutative Lie algebra with a Lie bialgebra structure. Example 3.3.7 Consider the Lie algebra (g, [·, ·]) over a field k of characteristic zero with basis elements X, Y ∈ g that satisfy [X, Y ] = X. Then define We want to prove that this is an r-matrix on (g, [·, ·]). The skew-symmetry is evident. Moreover r is a solution of the classical Yang-Baxter equation on g: CYB(r) = [r 12 , r 13 ] + [r 12 , r 23 ] + [r 13 , r 23 ] We recover the Lie bialgebra structure γ given in Example 3.2.5 because i.e. γ = δr. Thus (g, [·, ·], r) is a triangular Lie bialgebra, r is a triangular structure on (g, [·, ·], γ) and a r-matrix on g. Remark 3.3.8 If φ : g → a is a Lie algebra homomorphism to another Lie algebra a and r a triangular structures on g, then (φ ⊗ φ)r is a triangular structure on a. We prove this in Lemma 6.3.1. This is not true for an arbitrary coboundary structure. For example the image of a quasitriangular structure under a Lie algebra homomorphism does not need to be a quasitriangular structure (see e.g. [39, page 29]).
As promised before we pass from the infinitesimal world of Lie bialgebras to the global world of Poisson-Lie Groups, inspired by Theorem 3.2.8. Proposition 3.3.9 Let r ∈ g ∧ g be an r-matrix on g. The elements π λ r and π ρ r defined for x ∈ g by π λ r (x) = (T e λ x ⊗ T e λ x )r is a Poisson-Lie structure on G and it coincides with the one obtained in Theorem 3.2.8.

The Etingof-Schiffmann Subalgebra
In this section we discuss non-degeneracy of r-matrices. Remark that there is a one-to-one correspondence between non-degenerate r-matrices on g and non-degenerate 2-cocycles on g  Proof: We follow [39, Proposition 5.2]. Assume there is a non-degenerate r-matrix r ∈ Λ 2 g and consider the corresponding symplectic form ω ∈ Λ 2 g * . Now g is semisimple, then by Whitehead's Lemma (see Theorem 3.1.4) one has H 2 (g, C) = {0}, i.e. we find a 1-cocylce f ∈ g * such that for any x, y ∈ g. On the other hand, the Killing form κ ∈ 2 g * is non-degenerate on any semisimple Lie algebra. One can define a flat map with respect to κ with inverse map : g * → g. Let x, y ∈ g be arbitrary and define z = −f . Then where we use the associativity of κ (see Proposition B.2.2 ii.)) in ( * ). If we set y = z one has ω(x, z) = κ(x, [z, z]) = 0 for all x ∈ g and ω is degenerate, which is a contradiction.
However, one can always find a Lie subalgebra h ⊆ g such that a r-matrix on g is nondegenerate viewed as an element of h ∧ h. For this we need the classical Yang-Baxter equation in coordinate form: let g be a finite-dimensional Lie algebra with basis e 1 , . . . , e n ∈ g. Since [e i , e j ] ∈ g there are unique numbers C k ij ∈ k, called the structure constants, such that r ij e i ∧ e j ∈ g ∧ g be a r-matrix. (r jm r kl C i jk + r ij r mk C l jk + r ij r kl C m jk ) = 0, (3.4.4) where i, m, l ∈ {1, . . . , n}.
Proof: Since r ij = −r ji ∈ k one has r = n i,j=1 r ij e i ⊗ e j . Then (r jm r kl C i jk + r ij r mk C l jk + r ij r kl C m jk )e i ⊗ e m ⊗ e l and the classical Yang-Baxter equation CYB(r) = 0 is equivalent to n j,k=1 (r jm r kl C i jk + r ij r mk C l jk + r ij r kl C m jk ) = 0 for all i, m, l ∈ {1, . . . , n}.
This result allows the following Proposition 3.4.3 Let r ∈ g ∧ g be a r-matrix. Then the set is a Lie subalgebra of g and r ∈ h r ∧ h r ⊆ g ∧ g is non-degenerate in h r ∧ h r .
Proof: We follow [39, Section 3.5]. The isomorphism k ⊗ g ∼ = g implies h r ⊆ g. Moreover, h r is a Lie subalgebra of g since, if we take f, g ∈ g * , then The next step is to show that r is an element of h r ∧ h r . Since we already proved that h r ⊆ g is a Lie subalgebra we can choose a basis e 1 , . . . , e k ∈ g of h r , where k ∈ {1, . . . , n}, and complete this to a basis e 1 , . . . , e n ∈ g of g. Thus h r = span k {e 1 , . . . , e k }. For an arbitrary f ∈ g * this implies (f ⊗ 1)r =  Since f ∈ g * was arbitrary there must be r ij = 0 for j / ∈ {1, . . . , k}. By the skew-symmetry of r ij this implies r ij = 0 for i / ∈ {1, . . . , k} or j / ∈ {1, . . . , k}. Then indeed r ij e i ∧ e j ∈ h r ∧ h r .  In other words, the matrix (r ij ) ij ∈ M k×k (k) has full rank. For this r is non-degenerate viewed as an element of h r ∧ h r .
Following a discussion [94] with J. Schnitzer, we do the following Definition 3.4.4 (Etingof-Schiffmann Subalgebra) The Lie subalgebra h r defined in Eq.
(3.4.5) is called Etingof-Schiffmann subalgebra for a r-matrix r ∈ g ∧ g. The corresponding connected Lie group H r is said to be the Etingof-Schiffmann subgroup for r.
In Chapter 6 this is our tool to produce obstructions to twist star products on the sphere S 2 .

Twist Deformation
While the last chapter was dedicated to the understanding of Lie algebras with additional structure, also with some geometric aspects, we start by discussing pure algebraic features of general algebras here. The ideas and notions of Hopf algebras are developed in Appendix A and appear here the first time. A introduction to this topic is given in [70,Chapter 1]. Also consider [27], [72] and [100,Chapter 4]. In a way, Hopf algebras take over the role of the Lie group, like the algebra does for the manifold. By this we mean the following: in Chapter 2 we talked about Lie groups acting on manifolds and some of the involved properties. Now we define a process such that a Hopf algebra can act on an algebra. Of course there is no smooth structure. Instead, the algebraic relations of a Lie group action are taken as the new axioms. In this case the algebra is said to be a (left) Hopf algebra module. For our purpose this is still not enough, thus we force the module structure to be compatible with the algebra multiplication in addition. This leads to (left) Hopf algebra module algebras. But here comes the interesting question: what happens if we deform the Hopf algebra? Is this possible in a way such that the deformed Hopf algebra is still a Hopf algebra? And is there even a natural way to deform any (left) module algebra of this Hopf algebra such that the deformed algebra is a (left) module algebra for the deformed Hopf algebra? For a general deformation there is no reason for this to be the case, but for twist deformations there are positive answers to all of these questions and we give them in Section 4.1. We start by defining twists on a Hopf algebra and prove that they deform the Hopf algebra and any (left) module algebra of this Hopf algebra in the way we asked for. In a parallel move we evolve this theory for the inverse of a twist. This might sound exaggerated and quite trivial, but there are two reasons why we insist on this: first, the defining properties and deformations change slightly, thus, one has to be careful. And second, in literature both notions occur and it is reasonable to understand them both. As a last word to this topic we also prove that both definitions and the resulting deformations are indeed equivalent.
Since we are interested in star products we have to modify our definition of twist, i.e. we have to define twists on formal power series of universal enveloping algebras. Section 4.2 starts with a short repetition on universal enveloping algebras, their Hopf algebra structure and formal power series. We also discuss the problem that the resulting object is no Hopf algebra anymore in general. In this special situation we want to apply the results of Section 4.1. Instead of Hopf algebra twists we take those of the formal power series of universal enveloping algebras and instead of arbitrary algebras we take the algebra of smooth real-valued functions on a Poisson manifold with pointwise multiplication. If the deformed multiplication on the smooth functions is a star product on the manifold, this star product is said to be a twist star product. Moreover, one says that an arbitrary star product on a Poisson manifold can be induced by a twist if there is a twist on the formal power series of any universal enveloping algebra such that the deformation of the pointwise multiplication via this twist gives the star product. Thus at the end of this chapter we arrive at our objects of interest: star products and in particular the question whether they can or can not be induced by a twist. As a last observation we prove that a twist on the formal power series of a universal enveloping algebra always induces a r-matrix on the underlying Lie algebra. We stress that this is no one-to-one correspondence. This is the most important connection to Chapter 3. In the next chapter we finally see the connection to homogeneous spaces.

Twists and left Hopf Algebra Module Algebras
We need the notion of Hopf algebras and refer to Appendix A for a conceptional motivation of the involved axioms (see Definition A.0. 19). For instance, we use Sweedler's notation for the coproduct of a Hopf algebra element ξ, which is also discussed in detail in this appendix (consider Remark A.0.13). Do not mix this up with the short notation F = F 1 ⊗ F 2 of an arbitrary element F of the tensor product of a Hopf algebra. Let (H, +, ·, η, ∆, , S, k) be a Hopf algebra and F ∈ H ⊗ H. We introduce some very useful abbreviations

All of them are elements in
The first condition is said to be the 2-cocycle condition and the second one the normalization property.
Remark 4.1.2 There is also a kind of 2-cocylce condition for F −1 , since F is invertible. It is equivalent to Definition 4.1.1 i.), since we know that inverting reverses the order of elements and that the inverse element of (∆ ⊗ 1)F in H ⊗3 is (∆ ⊗ 1)F −1 and so on. Furthermore, F −1 fulfils the normalization property  We split the proof into three parts. First we prove that (H, +, ∆ F , , k) is a coalgebra (see Definition A.0.12). Since the algebra structure is not changed the second step is to check if the algebra and the coalgebra structure are compatible in the sense of a bialgebra (consider Definition A.0.17). Finally, S F has to be an antipode to finish the proof. Thus the first question is whether the diagrams (A.0.29) commute. To prove that ∆ F satisfies the coassociativity condition we need the coassociativity of ∆, the 2-cocycle condition of F and the 2-cocycle condition of F −1 . Furthermore, we use that ∆ is an algebra map. Let ξ ∈ H. Then Then the normalization property of F and F −1 together with the fact that is an algebra map implies the second (and third) coalgebra axiom: for ξ ∈ H we calculate This means that (H, +, ∆ F , , k) is a coalgebra and the first step is done. According to Definition A.0.17 (H, +, ·, η, ∆ F , , k) is a bialgebra if ∆ F and are algebra maps. Thus it remains to check this property for ∆ F . Let ξ, ζ ∈ H. Then since ∆ is an algebra map. Also ∆ F (1 H ) = F · ∆(1 H ) · F −1 = 1 H ⊗ 1 H holds. The third and last step is to check that S F is an antipode of the bialgebra (H, +, ·, η, ∆ F , , k). First of all, remark where we used several times the normalization property of F −1 and the antipode axiom for S and ∆ as well as the 2-cocylce condition for F −1 . Similarly Now the antipode axiom for S F and ∆ F is not hard to verify. Let ξ ∈ H. Then again by the antialgebra map property and antipode property of S and similarly Then S F is an antipode compatible to the bialgebra structure and (H, +, ·, η, ∆ F , , S F , k) is a Hopf algebra.
Assume that (H, +, ·, η, ∆, , S, k) is a cocommutative Hopf algebra, i.e. for ξ ∈ H one has or, equivalently, the commutativity of the following diagram where σ : H ⊗ H → H ⊗ H denotes the isomorphism flipping the tensor components called the braiding isomorphism. The deformed coproduct is not cocommutative in general, because for ξ ∈ H one has which in general does not coincide with In this sense we can say that the twist F sometimes also induces a quantization of the Hopf algebra.
and ( ⊗ 1)F −1 = (1 ⊗ )F −1 = 1 H . Thus F −1 is a twist on (H, +, ·, η, ∆ F , , S F , k) if F is a twist on (H, +, ·, η, ∆, , S, k). Arranging the above equations in another order we see that also the converse is true. Then F is a twist on (H, +, ·, η, ∆, , S, k) if and only if F −1 is a twist on (H, +, ·, η, ∆ F , , S F , k) and the convention chosen in Definition 4.1.1 is equivalent to the axiom system where we defined J = F −1 . The procedure of twisting is even involutive, i.e. twisting the twisted structure gives back the original one. This can be seen quite easyly, since for ξ ∈ H one has and This is called twisting back. Thus it is not surprising that both conventions appear in the literature. We want to follow the F-convention as it is done in [6], but remark that the Jconvention is also very common. Consider for example the inspiring book [39] by P. Etingof and O. Schiffman.
In a next step we show that a twist F on (H, +, ·, η, ∆, , S, k) also induces a deformation where we used the 2-cocycle property (4.1.2) of F −1 and the left H-module properties (4.1.10) and (4.1.12) of A several times. This shows that the product is associative. There is even the same unit element 1 A for . To see this, take a ∈ A and use properties (4.1.11), (4.1.13) and the normalization property (4.1.3) of F −1 to get Remark that also the k-linearity of was needed to prove the above formula. This is enough to show that A F is an algebra, since is linear. The only problematic property of Definition 4.1.5 we have to check is (4.1.12). Let a, b ∈ A and ξ ∈ H. Then where we use the short notation ∆ F (ξ) = ξ F (1) ⊗ ξ F (2) . We applied Eq. This theorem tells us that we can deform the product of a left H-module algebra to obtain a left module algebra of the deformed Hopf algebra. Assume that (A , ·, +, k) is a commutative left H-module algebra, i.e. ab = ba for all a, b ∈ A . Then the deformed algebra (A , , +, k) (which we may shortly denote by A F as in the last proof) is not commutative again in general, since for a, b ∈ A the term Thus F will not only deform but also quantize A in a way which is compatible to the deformation (quantization) of H.

Twist Star Products
We are only interested in the case when H is a universal enveloping algebra U(g) of a (finitedimensional) Lie algebra g over k. As a short repetition, the universal enveloping algebra of g is the tensor algebra of g modulo the relation for all ξ, ζ ∈ g. One structures U(g) as a Hopf algebra by setting where ξ ∈ g. This extends to algebra (anti) homomorphisms and yields a cocommutative Hopf algebra. In a general Hopf algebra H one calls an element ξ ∈ H that satisfies ∆(ξ) = ξ ⊗ 1 H + 1 H ⊗ ξ primitive. We want to pass to formal power series thus we have to recap them first.
Let k be a field of characteristic zero and define K = k[[ ]] to be the space of all sequences a = (a 0 , a 1 , a 2  where (a 1 , a 2 , ...) is an arbitrary element of K such that a n is the smallest coefficient that is not equal to zero. For the sequence (0, 0, ...) that is constant zero one defines the valuation to be 0. One can check that this is indeed a well-defined valuation on K called the -adic valuation. The topology induced by the -adic valuation is said to be the -adic topology and one can check that K is complete with respect to this topology. Moreover, if we extend the valuation to V    (4.1.14) by Theorem 4.1.6. Axiom iii.) assures that coincides with the original multiplication · on A in zero order of . If we think of quantization this is the classical limit that gives back the classical structure.
The main motivation to consider twist products , defined in Eq. (4.1.14), is that in many cases denotes a star product. We do not give a motivated introduction into the rich field of star products and formal deformation quantization, but refer to [30], [37,Chapter 3] and [105,Chapter 6]. For this thesis it suffices to see star products as products on formal power series of smooth functions of a manifold. We are mainly interested in their existence but not in their properties. Some nice examples of star products can be found in [85]. Remark that Kontsevich proved that there is a star product on any Poisson manifold (c.f. [61]).

Definition 4.2.3 (Star Product) A star product on a Poisson manifold (M, π) is a R[[ ]]-
bilinear map (4.2.7) of the form r C r (f, g), Condition ii.) of Definition 4.2.3 guarantees, that the classical limit → 0 gives back the commutative pointwise multiplication of functions, while condition iii.) can be interpreted as follows: the star product deforms the Poisson bracket {·, ·} (or equivalently the Poisson bivector π corresponding to {·, ·}). This is the correspondence principle we mentioned in the introduction. We only omitted a prefactor. If is a star product on a symplectic manifold (M, ω) one says that deforms the symplectic structure ω of M . This is nearby thinking of Eq. (3.2.23).
Inspired by Theorem 4.1.6 we have the following where m denotes the multiplication There is a connection between twists on U(g)[[ ]] and r-matrices on g. The correspondence principle states that the first order of the commutator of the star product gives the Poisson bracket of the manifold. In a very similar way a twist can be seen as a deformation of a r-matrix. Conversely, the skew-symmetrization of the first order of a twist gives a r-matrix. This is part of the following Proposition 4.2.5 Let g be a Lie algebra over R and Then is a r-matrix on g, where σ denotes the braiding isomorphism.
Proof: According to Definition 4.2.1 iii.

r-Matrices that lead to Homogeneous Structures
In this chapter we prove one of the main results of the thesis. Here we connect all previous chapters and justify their study. In a nutshell we show the following: if the Poisson bivector of a symplectic manifold is the image of a r-matrix under a Lie algebra action, the manifold has to be a homogeneous space. It is necessary to demand that the involved Lie algebra action integrates to a Lie group action. The manifolds we later consider are compact thus the integration condition is fulfilled naturally. Then an important step is to argue that this Lie group action is already locally transitive in this situation. Here the symplectic character of the manifold is essential. Remark that we develop parallel to the proceeding analogous results for the Etingof-Schiffmann subgroup of the r-matrix. It is surprising as well as of enormous use that one can restrict the Lie algebra to the Lie subalgebra in which the r-matrix is non-degenerate and still gets the desired result. Thinking of Section 2.3 and Section 2.4 there are many candidates for obstructions now: the higher pretzel surfaces are not homogeneous spaces and the 2-sphere can only be structured as a homogeneous space in very few ways. The question is whether all the requested assumptions are given for these examples. And moreover: why should it be desirable to find obstructions to Poisson bivectors that are images of r-matrices through Lie algebra homomorphisms? While Section 5.1 is fully dedicated to the proof of the essential theorem, Section 5.2 answers the latest question. The main motivation to consider such Poisson bivectors is the following: they occur naturally in the presence of twist star products. We have already shown in Proposition 4.2.5 that every twist on the formal power series of a universal enveloping algebra induces a r-matrix on the corresponding Lie algebra. If the twist star product deforms the symplectic structure of the manifold, the corresponding Poisson bivector is of the requested form. Thus we do not only give obstructions to special Poisson bivectors but to twist star products. Then this chapter ends with the statement that any connected compact symplectic manifold endowed with a twist star product can be structured as a homogeneous space. Following the way of the Etingof-Schiffmann subgroup we even know that there is a Lie group that acts transitively on the symplectic manifold such that the r-matrix is non-degenerate in the corresponding Lie algebra.
i.e. the restriction φ hr | p of φ| p to h r still satisfies the transitivity condition on T p M for every p ∈ M . Since φ was integrable this is also true for the restriction to a Lie subgroup. This gives Φ| Hr : H r × M → M , the restriction of Φ to H r × M , which is locally transitive and we can restrict ourselves without loss of generality to the Etingof-Schiffmann subalgebra in which r is non-degenerate.
Our aim is to show that the setting of the last lemma already implies that M is a homogeneous space. For this we need another Proof: Let φ be the Lie algebra action corresponding to the locally transitive Lie group action Φ. Fix p ∈ M and denote by g p the Lie algebra corresponding to G p . Remark that this is possible since G p ⊆ G is a Lie group according to Theorem 2.1.1 ii.). The map β p : G/G p g · G p → Φ(g, p) ∈ M is a diffeomorphism if Φ is transitive according to Theorem 2.2.6. Our aim is to show that β p is also a diffeomorphism if Φ is just locally transitive. We first prove that β p is a local diffeomorphism. We have already shown in Theorem 2.2.6 that β p is well-defined and smooth. Thus we can differentiate β p at e · G p ∈ G/G p and obtain T e·Gp β p : g/g p → T βp(e·Gp) M = T p M. (5.1.8) Then the mapφ Since β p (exp(0 · ξ) · G p ) = Φ(e, p) = p the last calculation implies that β p (exp(tξ) · G p ) = p for all t ∈ R. This shows exp(tξ) ∈ G p for all t ∈ R and for this ξ ∈ g p , thus [ξ] = 0 andφ is injective.
Then for any g ∈ G 1.11) where g −1 : G/G p g · G p → (g −1 g ) · G p ∈ G/G p . Since Φ g and −1 g are diffeomorphisms also T g·Gp β p is bijective. Thus β p is a local diffeomorphism for all p ∈ M , i.e. for any g · G p ∈ G/G p there is an open neighbourhood U ⊆ G/G p of g · G p such that β p (U ) ⊆ M is open. Choose any point q ∈ G · p of the orbit of p under Φ. Then for every g ∈ Φ −1 p ({q}) ⊆ G there is an open neighbourhood U ⊆ G/G p such that β p (U ) ⊆ G · p ⊆ M is open. Since q ∈ β p (U ) we have found an open neighbourhood of q that is contained in G · p and the orbit G · p is open. Let again p ∈ M be arbitrary and assume that there is a point q ∈ M \ {G · p}. Since M is always partitioned by the orbits of any group action one has G · p ∩ G · q = ∅, since otherwise G · p = G · q what is not possible because q / ∈ G · p. By this we get a partition of M by open sets what gives a contradiction if M is connected. This concludes the proof. Now we just collect our results and conclude one of the main theorems of this thesis.
Theorem 5.1.3 [15]. Let (M, ω) be a connected symplectic manifold with corresponding Poisson bivector π, r ∈ g ∧ g a r-matrix and φ : g → Γ ∞ (T M ) a Lie algebra action of g on M that integrates to a Lie group action Φ : G × M → M of G on M such that for any p ∈ M one has Then M can be structured as a homogeneous space via Φ. Moreover, the Etingof-Schiffmann subgroup H r acts transitively on M .
Proof: This is a direct consequence of Lemma 5.1.1 and Lemma 5.1.2.
For this, a geometric information provides topological impact on the manifold: if the Poisson bivector is the image of a r-matrix under an integrable Lie algebra action, the manifold has to be a homogeneous space. For instance, the Euler characteristic of such a compact space has to be non-negative according to Theorem 2.3.16. We make use of this in the next chapter to get obstructions on the higher pretzel surfaces. The great benefit of the second statement of Theorem 5.1.3 is that the r-matrix is non-degenerate in the Lie algebra of a Lie group that acts transitively on M . This means we have more information on an infinitesimal level. Later on this helps us to find obstructions in the case of M = S 2 .
It is clear that one can apply Theorem 5.1.3 to any leaf of a symplectic foliation of a Poisson manifold. Remember that for any Poisson manifold (M, π) the image of the sharp map is a smooth distribution. By this we mean that im( ) is a set of linear subspaces im( x ) of T x M , where x ∈ M , such that each is spanned by a finite number of smooth vector fields X 1 , ..., X k ∈ im( ), i.e. im( x ) = span{X 1 (x), ..., X k (x)}. One can also imagine im( x ) as the set 1.14) i.e. the set of tangent vectors which are images of Hamiltonian vector fields at x. Now im( ) integrates to a foliation of M , that is a cover of M by connected subsets which are said to be the leaves of this foliation. Moreover, the Poisson bivector π induces symplectic Poisson bivectors on these leaves and for this we obtain a foliation of M called a symplectic foliation. The leaves are maximal symplectic submanifolds of M which are said to be the symplectic leaves of M . All these facts can be found in [101,Chapter 2]. Thus we can modify our last result in the following way: Corollary 5.1.4 Let (M, π) be a Poisson manifold. Any symplectic leaf (S, π S ) of M whose Poisson bivector is the image of a r-matrix under an integrable Lie algebra action φ : g → Γ ∞ (T S) is a homogeneous space. Moreover, the Etingof-Schiffmann subgroup of this r-matrix acts transitively on S.

Structures induced by Twist Star Products
As an a posteriori motivation to consider Poisson bivectors of the form (5.1.3) we examine star products that are induced by a twist. In Corollary 4.2.6 we have seen the connection between twist star products and r-matrices. It is also interesting to consider the Poisson bracket and the Poisson bivector which are deformed by a twist star product.
ii.) the Poisson bracket reads iii.) there is a Lie algebra action φ of g on M defined for any ξ ∈ g by

2.3)
where L denotes the Lie derivative, iv.) the Poisson bivector π corresponding to {·, ·} reads for any p ∈ M , where e 1 , . . . , e n is a basis of g and r = 1 2 n i,j=1 r ij e i ∧ e j . Proof: In particular, Then the corresponding Poisson bracket satisfies for all f, since pointwise multiplication is commutative, where r is the r-matrix on g defined in Eq.
φ is a Lie algebra anti-homomorphism. This proves the third part. If we assume g to be finite-dimensional with basis e 1 , . . . , e n ∈ g one has  Proof: Since any flow of a vector field on a compact manifold is complete, a theorem of Palais (c.f. [71,Theorem 6.5]) implies that any Lie algebra action on a connected compact manifold integrates to a Lie group action.

Chapter 6
Obstructions and Examples of Twist Star Products In this chapter we gather all previous notions and results to prove or disprove the existence of twist star products deforming the symplectic structure on connected compact manifolds of dimension 2. One can generalize these results easily to arbitrary Poisson manifolds that inherit symplectic leaves of these types. It turns out that the examples that allow existence of twist star products deforming the symplectic structure are quite rare. In fact there is only one example of such a twist star product on the 2-torus and obstructions in all other cases.
is a non-degenerate 2-form on M the corresponding Liouville form symplectic manifold is diffeomorphic to T(g) for a unique g ∈ N 0 . We are also able to construct a generic symplectic form on T(g). Since T(g) is a manifold it is Hausdorff and second countable. Together with the property of being connected this implies the existence of a Riemannian metric m ∈ Γ ∞ (S 2 T * T(g)) on T(g), i.e. for any p ∈ T(g) is a positive definite symmetric bilinear form on T p T(g) (c.f. [2,Proposition 5.5.11]). In a positively oriented local chart (U, x = (x 1 , x 2 )) of T(g) one has Then define which gives a 2-form ω ∈ Γ ∞ (Λ 2 T * T(g)) on T(g) by gluing together (6.0.4) on an atlas of T(g).

The higher Pretzel Surfaces
According to [66,Section 12.1.4] the Euler characteristic of T(g) is for any g ∈ N 0 . In particular, one has χ(T(g)) < 0 for g > 1. There is no twist star product on T(g) deforming a symplectic structure of T(g).
On the other hand, there is a star product on any T(g) obtained by the Fedosov construction since T(g) is symplectic (c.f. [40]). Thus there are star products deforming the symplectic structure on T(g) which can not be induced by a twist and we have infinitely many examples. This gives a partial proof of the main theorem stated in the introduction.

The 2-Torus
Surprisingly, there is a twist star product on the 2-torus T 2 . Consider R 2 with coordinates (x 1 , x 2 ). If we choose the zero Lie algebra g over R 2 then U(g) = S • R 2 can be endowed with a Hopf algebra structure ∆(x) = x ⊗ 1 + 1 ⊗ x, (x) = 0 and S(x) = −x for all x ∈ R 2 . As described in Section 4.2 this can be extended to a Hopf algebra is the well-known Weyl-Moyal star product on . It deforms the symplectic structure on T 2 . More information can be found in [75] and [77].

The 2-Sphere
Finally, we consider T(0), i.e. the sphere S 2 . It is also a connected compact symplectic manifold: take for example the symplectic form ω ∈ Γ ∞ (Λ 2 T * S 2 ) defined for any x ∈ S 2 and v, w ∈ T x S 2 by where · denotes the standard scalar product on R 3 and × the vector product. While the closedness of ω is trivially fulfilled, also the non-degeneracy can be concluded quite easily since T x S 2 is the plain of vectors that are perpendicular to x in R 3 . Unfortunately, the trick used in Theorem 6.1.1 does not apply here since χ(S 2 ) = 2 > 0. But we have seen in Section 2.4 that there are only very few connected (non-equivalent) Lie groups that act transitively and effectively on S 2 .
The question is now the following: are the additional assumptions of being connected and effectiveness to specific to produce general obstructions? Anyway, the connected Lie groups that act transitively and effectively give only a few possibilities to structure S 2 as a homogeneous space. But luckily for any transitive action on S 2 one can construct a connected one that acts effective in addition, such that the r-matrix survives the transformation. As an immediate consequence of Corollary 5.2.3 we can assume that the Lie group that acts transitively is connected: if there is a twist star product on a compact symplectic manifold the Etingof-Schiffmann subgroup (a connected Lie group) corresponding to the r-matrix induced by the twist acts transitively on the manifold and the r-matrix is non-degenerate in the corresponding Lie algebra. We are interested in obstructions, thus this is exactly the implication we want since the existence of this connected Lie group acting transitively on S 2 produces a contradiction in the end.
We now prove that r-matrices fit well in our categorial frame.
Proof: This is a simple calculation: by the Lie algebra homomorphism property of φ. Then, if r is a r-matrix one has CYB(r) = (φ ⊗ φ ⊗ φ)0 = 0. Also the skew-symmetry ofr is clear since φ is linear.
We pass from transitive to transitive and effective actions. With the help of the last lemma we do not loose the r-matrix. Proposition 6.3.2 Let G be a Lie group acting on a manifold M via Φ and define If r is a r-matrix on the corresponding Lie algebra g then there is a r-matrix on the Lie algebra g/n corresponding to G/N . If the action Φ : G × M → M is transitive, so is the effective action Proof: We already know that N is a Lie subgroup of G. To see that it is a normal subgroup choose g ∈ G and n ∈ N and calculate which shows that gng −1 ∈ N . Thus gN g −1 ⊆ N for all g ∈ G and N is normal. It is well-known (c.f. [91,Proposition 5.7.8]) that the quotient G/H of a Lie group G and a normal subgroup H ⊆ G is again a Lie group and there is a Lie group homomorphism π : G → G/H. Consequently, we find a Lie group homomorphism π : G → G/N . Then its tangent map φ : g → g/n at e ∈ G is a Lie algebra homomorphism. Thus the first claim of the proposition follows from Lemma 6.3.1.
For the second one, we first check that Ψ defined in Eq. (6.3.3) is well-defined. Choose elements g, g ∈ G such that g · N = g · N , i.e. there is a h ∈ N such that g = gh. Then The smoothness as well as the action properties of Ψ are induced by Φ. It is effective by construction since the kernel N is eliminated.
Thus for our observations we can require without loss of generality that the Lie group acting transitively on the homogeneous space is connected and acts effectively in addition: if there is a r-matrix on g there also has to be one on g/n. For the sphere S 2 we have classified all such actions in Section 2.4 up the equivalence. Also this equivalence can be fixed.
x ∈ M . The Lie algebra action corresponding to φ maps a r-matrix on g to a r-matrix on g according to Lemma 6.3.1.
Finally, we can prove the main theorem stated in the introduction. Theorem 6.3.4 [15]. There is no twist star product on S 2 deforming any symplectic structure of S 2 .
Proof: Assume there is a twist star product = ∞ i=0 i C i (·, ·) on S 2 such that for all λ, µ ∈ k and v ∈ V , where 1 denotes the unit of (k, ·) and we set where 0 denotes the unit of (k, +) and 0 V the unit of (V, +). Moreover, these structures have to be compatible, i.e.
for all λ, µ ∈ k and v, w ∈ V . The map , called scalar multiplication, is omitted in most cases.
We do not want to give an introduction to linear algebra, but recall the definitions of the tensor product and the dual space of a vector space since they are used in this thesis. In the following k always denotes a field.
Example A.0.7 We start with the free abelian group on V × W . This is the set of finite strings of tuples in V × W modulo the relation on V ⊗ W , since for λ, µ ∈ k and a ⊗ b ∈ V ⊗ W one has by the linear extension and and thus the triple (V ⊗ W, +, k) is a k-vector space.
Remark that there is also a definition of tensor products via a universal property, consider e.g. [98]. As an example we want to discuss the k-vector space k ⊗ V for some k-vector space V and realize that it is isomorphic to V itself. This identification is necessary for the definitions of algebras and coalgebras.
Lemma A.0.8 Let V be a vector space over a field k. Then k ⊗ V ∼ = V ∼ = V ⊗ k.
Proof: The isomorphisms we are searching for are where the right hand side denotes the k-action on the k-vector space V . We prove that the map defined in Eq. (A.0.11) is an isomorphism. The proof for the second isomorphism works exactly the same. The map defined in Eq. (A.0.11) is surjective, since any v ∈ V can be produced via For injectivity assume there are λ ⊗ v, µ ⊗ w ∈ k ⊗ V such that λv = µw. If λ = 0 then µw = 0, i.e. µ = 0 or w = 0 and for this µ ⊗ w = 0 = λ ⊗ v because Thus assume that λ = 0. Then v = λ −1 µw and This completes the proof.
Now we discuss the last vector space example.
Example A.0.9 The set V * of all k-linear maps V → k is called the dual vector space of the k-vector space (V, +, k). Thus an element φ ∈ V * is a map φ : V → k such that for all λ, µ ∈ k and v, w ∈ V φ(λv + µw) = λφ(v) + µφ(w). (A.0.13) Then V * becomes a k-vector space structure if we define an addition on V * pointwise for two elements φ, ψ ∈ V * by for all v ∈ V , where the right hand side denotes the addition in k and if we define an action of k on V * pointwise for λ ∈ k and φ ∈ V * by where the argument of φ in the right hand side is the k-action of λ on v. The vector space axioms are easy to check. Note that there is a way to identify an element v ∈ V with an element in the Moreover, if φ ∈ V * and ψ ∈ W * we define a map φ⊗ψ ∈ V * ⊗W * by (φ⊗ψ)(v⊗w) = φ(v)⊗ψ(w) for v ∈ V, w ∈ W . Remark that by Lemma A.0.8 one has φ(v) ⊗ ψ(w) ∈ k ⊗ k ∼ = k thus φ ⊗ ψ : V ⊗ W → k. We calculate for λ, µ ∈ k and v ⊗ w, We are now able to define a very central object in Hopf algebra theory, first in terms of rings, fields and actions: an algebra.
where 1 : A → A denotes the identity map on A , i.e. · has to be associative and there has to be an element 1 A ∈ A such that  We reformulated the definition of an algebra via the commuting diagrams (A.0.26) because we want to dualize our objects and notions. The proceeding is now as follows: simply reverse arrows and demand objects to fill the gaps of the maps that do not fit any more in the diagrams.
Definition A.0.12 (Coalgebra) We call (C , +, ∆, , k) a coalgebra over a field k if there is a linear coproduct which is coassociative and a linear counit The coassociativity axiom and the axioms a counit has to satisfy are denoted in the commutativity of the three diagrams of (A.0.29), respectively.
The axioms of a coalgebra depicted in the commutativity of the diagrams (A.0.29) read in formulas where 1 : C → C denotes the identity on C .
Remark A.0.13 One could wonder what part of ∆(c) ∈ C ⊗ C for a c ∈ C is applied to ∆, or 1 in Eq. (A.0.30) and Eq. (A.0.31), respectively. We want to discuss this problem. ∆(c) ∈ C ⊗ C is a linear combination of elements for a finite number n ∈ N. Having this in mind we use the short notation (called Sweedler's notation) More general we define elements of the form c (1) ⊗ c (2) ⊗ · · · ⊗ c (r) ∈ C ⊗ · · · ⊗ C for any r ∈ N and c ∈ C by . . .

• ∆(c).
For r = 1 and r = 2 we get back the definitions in Eq. (A.0.32) and Eq. (A.0.35), respectively. As in this two cases also permutations of ∆ and 1 in the defining equations lead to c (1) ⊗c (2) ⊗· · ·⊗c (r) as a consequence of the coassociativity of ∆. Let us prove this for r = 3.
where we used the coassociativity in the fourth equation for the element c (1) . Similarly for the other permutations. By induction this result is true for any r ∈ N. As another short notation that we sometimes use, define where we set ∆ 0 = ∆. While ∆ adds another C , the counit reduces its argument about one C . More precise, We prove this again only for r = 3.
where we used the counit axiom for the element c where 1 A and 1 B denote the units of A and B, respectively. A k-linear map f : C → D between two coalgebras C and D over k is said to be a coalgebra map if it satisfies Remark A.0.15 One can structure the tensor product of two (co)algebras as a (co)algebra again. Explicitly, we define for two algebras A and B a product on the vector space A ⊗ B by for all a, b ∈ A and c, d ∈ B and extend this linearly to A ⊗ B. Because of the associativity of the product on A and B, respectively, we get for a, b, x ∈ A , c, d, y ∈ B which is the associativity of ·. The unit of A ⊗ B is the tensor product of the units on A and B, i.e. 1 A ⊗B = 1 A ⊗ 1 B which can be seen via for all a ∈ A , b ∈ B. For two coalgebras C and D we define the coproduct on C ⊗ D for all c ∈ C , d ∈ D by and extend this linearly to C ⊗ D. We used again the notation introduced in Eq. (A.0.32). The coassociativity is satisfied because for c ∈ C and d ∈ D one has by the coassocitivity in C and D, respectively. The 1 in the equation above denotes the identity map on C ⊗ D. The counit of C ⊗ D is the tensor product of the counits of C and D, i.e.
where the last equation follows from the counit property on C and D, respectively. The other counit axiom follows similarly.
The following should give an argument why on calls the process of passing from an algebra to a coalgebra (and back) dualizing. of the coproduct and counit, respectively. More explicit, one has for φ, ψ ∈ C * , c ∈ C and λ ∈ k The product · and the unit η make the dual space C * into an algebra. The unit element is . Conversely, if A is a finite-dimensional algebra, A * gets a coalgebra structure in the same fashion.
Proof: Assume that (H, +, ·, η, ∆, , k) is a bialgebra. It is just a matter of calculation to verify Eq. (A.0.49) and the equations following. Thus let g, h ∈ H and λ ∈ k.
where · in the third line denotes the product on the tensor product H ⊗ H defined in Eq. (A.0.39).
where the second equation holds since H is an algebra map, the third equation is due to one isomorphism of Lemma A.0.8 and the last equation is the definition of H⊗H .
where the equation next to the last follows from the fact that ∆ H is k-linear and again by an isomorphism of Lemma A.0.8. Finally, where the second equation follows because H is k-linear, the third equation holds because H is an algebra map and the last equation is the definition of k . Conversely, assume that equation (A.0.49) and the three equations following hold. The same equations of the first part of this proof reordered prove that ∆ and are algebra maps. Explicitly, we calculate for g, h ∈ H ∆ H (g · h) = ∆ H • ·(g ⊗ h) = (· ⊗ ·) • ∆ H⊗H (g ⊗ h) = (· ⊗ ·)(g (1) ⊗ h (1) ⊗ g (2) ⊗ h (2) ) = (g (1) · h (1) ) ⊗ (g (2) · h (2) ) = (g (1) ⊗ g (2) ) · (h (1) ⊗ h (2) ) where the second equation holds since · is a coalgebra map and we used again the multiplication on the tensor product H ⊗ H.
Remark that if H and G are Hopf algebras with antipodes S and S respectively, also the tensor product H ⊗ G is a Hopf algebra. The bialgebra structure is given by the usual tensor bialgebra structure, while the unique antipode is S ⊗ S . We did not yet define the corresponding morphisms:  c.) a proper ideal of g if it is an ideal of g but not equal to {0} or g.
b.) simple if g is not abelian and there is no proper ideal contained in g.
c.) semisimple if g can be written as a direct sum of simple Lie algebras.
d.) reductive if g can be written as a direct sum of simple and abelian Lie algebras.
It is clear from the definition that every proper ideal is an ideal and every ideal is a Lie subalgebra. where u(1) is a 1-dimensional vector space endowed with the Lie bracket that is identically zero. A more difficult approach is needed to classify reductive and for this semisimple and simple Lie algebras. This is called the Cartan-Killing Classification. For a proof that uses Dynkin diagrams consider [20].

B.2 Semisimple Lie Algebras
We start by defining a very central element in the theory of semisimple Lie algebras and follow [87], but also refer to [58,Chapter 2]. The eigenspaces of a Cartan involution of a Lie algebra provide a useful decomposition. For the definition we need the Killing form of a Lie algebra. Later we see that the non-degeneracy of this object characterizes complex semisimple Lie algebras. For a Lie algebra (g, [·, ·]) and an element x ∈ g we denote the adjoint endomorphism on g corresponding to x by ad(x) : g y → [x, y] ∈ g. is said to be the Killing form on g.
Already for an arbitrary Lie algebra κ has interesting properties: Proposition B.2.2 Consider a Lie algebra (g, [·, ·]) over a field k. Then the following statements hold.
i.) The map κ defined in Eq. (B.2.2) is a well-defined symmetric bilinear form.
ii.) κ is associative, i.e. for all x, y, z ∈ g the equation holds.
The last statement is a consequence of ii.). To prove this, take arbitrary elements x ∈ S and y ∈ g. Then [x, y] ∈ S because κ([x, y], z) = κ(x, [y, z] ∈g ) = 0 for all z ∈ g which proves the claim.
In particular, the last statement of this proposition is interesting for semisimple Lie algebras since there is always an ideal S in g, namely the radical of κ. If S = g the Lie algebra would be rather boring, thus one traceable claim to hope for a semisimple Lie algebra should be S = {0}. Since κ is symmetric according to i.) this is equivalent to the postulate that κ is non-degenerate in g. Our aim is to prove that this is not only a necessary but also a sufficient condition to characterize complex semisimple Lie algebras. We first prove that S controls all non-proper abelian ideals.
Lemma B.2.3 Let (g, [·, ·]) be a Lie algebra over a field k and J an abelian ideal of g. Then J ⊆ S.
Proof: Let x ∈ J and y ∈ g be arbitrary. Our first step is to show that the endomorphism ad(x)ad(y) (B.2.8) is nilpotent. More explicit, we prove (ad(x)ad(y)) 2 = 0. First of all (ad(x)ad(y))(z) ∈ J for all z ∈ g since For the proof of the next theorem we need another characterization of semisimple Lie algebras taken from [10, Theorem 6.5]: a Lie algebra is semisimple if and only if it has no non-zero abelian ideal.
Theorem B.2.4 Consider a finite-dimensional complex Lie algebra (g, [·, ·]). It is semisimple if and only if the Killing from κ is non-degenerate on g.
Proof: As argued before we prove that g has a non-zero abelian ideals if and only if S = {0}. If g has a non-zero abelian ideal J than {0} = J ⊆ S according to Lemma B.2.3. Assume conversely S = {0}. Then Cartan's Criterion (c.f. [1, Corollary 3.3.14]) implies that S is a proper ideal of g since κ(x, y) = 0 for all x, y ∈ S. Thus g is not semisimple.
In the following we often view a semisimple Lie algebra together with its Killing form. An object which is quite related to the Killing form is a Cartan involution.
One could ask in which situation such a Cartan involution exists and when it is unique. There is an answer for real semisimple Lie algebras: Theorem B.2.6 There is a Cartan involution on a real semisimple Lie algebra which is unique up to inner automorphisms.
A proof can be found in [58,Theorem 3]. It uses Theorem B.2.4 and the existence of a compact real form of a finite-dimensional semisimple Lie algebra. We want to say some words to the last point. A real Lie algebra g 0 is said to be a real form of a complex Lie algebra g if one has g ∼ = C ⊗ R g 0 .

(B.2.12)
Conversely, a complex Lie algebra g is said to be a complexification of a real Lie algebra g 0 if Eq. (B.2.12) holds. A famous theorem of Cartan says that if g is a complex semisimple Lie algebra there is always a real form g 0 of g and g 0 is a compact Lie algebra (c.f. [89]). Now consider a real semisimple Lie algebra g with a Cartan involution Θ. Since it is an involution Θ has exactly two eigenvalues ±1. The corresponding eigenspaces are commonly denoted by k and p.

B.3 Root Space Decomposition and Iwasawa Decomposition
To get in touch with Iwasawa decompositions we use roots and Cartan subalgebras and follow [57,Chapter 3]. There are many equivalent definitions of Cartan subalgebras, but we characterize them via semisimple elements. Recall that for an element x of a Lie algebra (g, [·, ·]) the endomorphism ad(x) is diagonizable if there is a basis {x 1 , x 2 , . . .} of g such that [x, x k ] is proportional to x k for every k = 1, 2, . . .. Moreover, an abelian subalgebra h of g is said to be maximal abelian if there is no abelian subalgebra h of g such that h h g.
i.) An element x ∈ g is said to be semisimple if ad(x) is diagonizable.
ii.) If g is semisimple, a maximal abelian subalgebra h of g is said to be a Cartan subalgebra of g if it consists of semisimple elements.
Can one guarantee the existence of a Cartan subalgebra? This is discussed in the next The subset spanned by these elements is a Cartan subalgebra h of g. In general there are many Cartan subalgebras of g, but one can show that they have all the same dimension, which is said to be the rank of g.
Consider again a semisimple Lie algebra (g, [·, ·]) over an algebraically closed field k and  to be a weight system and α x i a weight of the adjoint representation of the Cartan subalgebra h. The elements α x i = 0 are said to be the roots of g and the set of roots ∆ is said to be a root system of g. We often omit the index x i . of g. It is called the root space decomposition of g corresponding to the Cartan subalgebra h. The number dim(g α ) is called the multiplicity of the root α.
Let us come back to the Cartan decomposition (B.2.13) of a semisimple Lie algebra. We consider a root space decomposition of the eigenspace p corresponding to the −1 eigenvalue of a Cartan involution Θ. The Cartan subalgebra of p shall now be denoted by a. We are not interested in all roots ∆ but only in the positive roots ∆ + . Choose a basis {α 1 , α 2 , . . .} of ∆. A root α ∈ ∆ is said to be positive if the first non-zero coefficient in the basis representation is positive. Then we define is said to be a Iwasawa decomposition of g. The Iwasawa decomposition at Lie group level is where G, K, A and N denote the connected Lie groups corresponding to g, k, a and n respectively.