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Quantum groups and Jordan structures

This paper is meant to be an informal introduction to Quantum Groups, starting from its origins and motivations until the recent developments. We call in particular the attention on the newly descovered relationship among quantum groups, integrable models and Jordan structures.


INTRODUCTION
Quantum groups represent at present one of the most important topics of mathematics and of mathematical physics.
The principal aim of the paper is to make the non-expert reader acquainted with this fast developing field and motivate the investigation towards the relationship with another important, though much earlier, branch of both mathematics and mathematical physics: that of Jordan structures. In the spirit of the word quantum, to denote a construction which includes the classical case in the appropriate limit, we would like to point out that a quantum analog of the profound link existing at the classical level between Lie and Jordan structures is worth exploring.
We take this opportunity to firstly give a short introduction to quantum groups. This is done in order to make our contribution understandable, and also to introduce are Conformal Field Theories which in turn describe string theories. Another strong physical motivation comes from Quantum Chromo-Dynamics (QCD): it has been argued that the scattering process at high energies occurs essentially in dimension 2. Moreover, to remain in the framework of quantum field theories, some of the characteristic features of Yang-Mills theories in dimension 4, like asymptotic freedom and dimensional transmutation, are also shared by the non-linear sigma model in dimension 2.
These are not the only reasons for working in low dimension. There is a branch of Physics in which low dimensional phenomena are observed and studied: Condensed Matter Physics. Superconductivity at high-temperature and quantum Hall effect are the most significant phenomena in this respect, but we may also cite technological applications like silicon chips and plastic films.
Beyond all this, low dimensional Physics has provided us with solvable models, among them the first complete examples of field theory with non-trivial interaction. This is by itself a good reason for such an investigation: models are so inherent to Physics! With this variety of interests in low dimensional Physics, there is no wander why a theory like QISM, which proved so powerful in yielding exact solutions for a broad class of 2-dimensional models, has attracted so much attention.
To understand how quantum groups are relaterd to QISM and what is the origin itself of the term "quantum group", due to Drinfeld [4], one should start from the fact that the QISM recovers the classical inverse scattering method (ISM) in the limit h → 0, h being the quantum parameter of the theory, while in the same limit the quantum group underlying QISM goes to the Poisson-Lie group underlying ISM. Let us briefly explain what is meant by this.
The ISM is about 25 years old and has been successful in solving certain types of classical field theories and Hamiltonian mechanics, like the ones formulated in terms of the sine-Gordon equation, the non-linear Schrödinger equation, the Korteweg de Vries equation, famous for its soliton solution and now fundamental in the matrix models of non-perturbative string theory (KdV hierarchy), the Toda systems, etc.
All these models are based on highly non-linear equations and the key of the ISM is to introduce an auxiliary space and new operators (Lax pairs) in terms of which one can write a linear equation equivalent to the original one. By doing so one finds the solution to a typical inverse problem analog to that of reconstructing a potential from the scattering data. This is achieved using the technique of the commuting transfer matrices, dating back to Onsager (since 1944). We do not enter into details, since this is beyond the scope of our paper, but what we have just mentioned is a crucial point in the history of quantum groups. The method of commuting tranfer matrices leads to solvability by building a number of conserved quantities in involution equal to the number of degrees of freedom, thus ensuring integrability by Liouville's theorem.
Here "in involution" means that all these integrals of motion, taken pairwise, have vanishing Poisson bracket; in the quantum case this translates to being mutually commuting. The main equation in the Hamiltonian formulation of ISM is (in the discrete case): where L is a Lax operator, that is a matrix -say N × N -depending on the classical fields, the curly bracket is the Poisson bracket, {L n (λ), L m (µ)} is a N 2 × N 2 -matrix with elements {L n (λ) ij , L m (µ) kl }, and r (commonly called classical r-matrix) satisfies the classical Yang-Baxter equation (YBE), leading quite directly to integrability: where, if r acts as a matrix on V ⊗ V , r αβ acts on V ⊗ V ⊗ V as r on the α-th and β-th components and as the identity on the remaining one.
If r(λ) ∈ End(V ⊗V ) and V is an N -dimensional complex vector space, then we can regard the classical YBE as being formulated in terms of the Lie algebra structure of End(V ). This suggests to recast ISM in a more general algebraic structure. Let G be a Lie algebra and r(λ) be a G ⊗ G -valued function, namely so that each term of the classical YBE belongs to G ⊗ G ⊗ G. The solution of the classical YBE, written in this way, is a universal solution, in the sense that for each yields a matrix solution of the classical YBE. For complex simple Lie algebras, the solutions of the classical YBE have been studied by Belavin and Drinfeld. In particular Drinfeld has shown that if G is a simple and simply connected Lie group, then G is a Poisson-Lie group, namely a Lie group with a compatible Poisson structure built on the functions defined on the group manifold, if and only if its Lie algebra is a Lie bialgebra (for the concept of bialgebra see below). We may say that the Lie bialgebras are the mathematical background for the theory of classical integrable models (which is essentially soliton theory in such a low dimension).
Remark 2.1. We like to point out that very recently Boldin, Safin and Sharipov [11] proved a surprising connection between Tzitzeica surfaces and ISM. The transformation that generates the family of such surfaces found by Tzitzeica [12] in 1910 and its slight generalizations obtained by Jonas [13,14] in 1921 and 1953 are known in the modern literature on integrable equations as Darboux or Bäcklund tranformations. They are used to construct the soliton solutions starting from some trivial solution of the equation u xy = e u − e −2u . It is worth mentioning that the paper [12] by Tzitzeica seems to be the first in the world where the equation u xy = e u − e −2u (the nearest relative of the sine-Gordon equation u xy = sinu) was considered.
Drinfeld's motivation for introducing quantum groups was to quantize the structure underlying classical integrable models and classical YBE in the sense of finding a quantum analog which recovers the classical theory in the limit for a quantum parameter going to zero. This program led him to quasi-triangular Hopf algebras in which a universal R-matrix is defined satisfying the quantum Yang-Baxter equation (QYBE) (see below for the definition) in each representation of the algebra. The goal of quantizing the whole structure rather then particular solutions of the classical YBE has thus been achieved.
The quantum R-matrix which in the limit goes to the classical r-matrix r(λ) ∈ G⊗G (solution of the classical YBE) lives in U q (G)⊗U q (G), U q (G) being a deformation of the universal enveloping algebra of G. After all U q (G) is the natural candidate for the purpose, but we would like to emphasize that one should not look only at the algebra aspect of the theory: the coalgebra structure plays a crucial role from both mathematical and physical standpoints [2] and both algebra and coalgebra are deformed in general.
Let V be a complex vector space and let R(λ), λ ∈ C, be an End C (V ⊗V )-valued function, then the equation is called Quantum Yang-Baxter Equation.
Comment. It is possible to consider more general dependence of the R-matrix on the spectral parameters and solutions have been found for different cases.
We now want to show that this equation is the key for the integrability of certain 1+1 dimensional quantum systems as well as two dimensional classical statistical systems (the equivalence between the two being one of the most interesting connections between different branches of low dimensional physics). 2) with no dependence on the parameters λ, µ, usually called spectral parameters. Also the equation with no parameter dependence is referred to in the literature as QYBE and to distinguish the two it is common to add "with spectral parameter" when referring to (2.2). In the QISM the spectral pa-rameter is needed for building up the generating functional for commuting conserved quantities.
For sake of simplicity, let us focus our attention on a vertex model of statistical mechanics. We consider a two dimensional square lattice with N × M points connected by bonds which can take n possible values. The partition function Z gives the probability attached to each configuration of the whole structure. In order to built it one starts by setting up the Boltzmann weights at each vertex. If the vertex has bonds ijkℓ then the weight is denoted by R i k j ℓ and can be arranged in a matrix form. In physical applications the R-matrix should satisfy certain conditions, like unitarity, symmetry, positivity (the same for 1+1 quantum field theories) and very often periodic boundary conditions are assumed and the limit N, M → ∞ is eventually taken.
Going back to the construction of the partition function, once the local properties of the model are set up, one looks for the probability for a certain configuration of N lattice points in a row. Assuming periodic boundary conditions this is given by the transfer matrix τ = tr( N i=1 R (i) ) (the trace appearing because of periodicity). Finally the partion function is obtained by repeating the procedure on the M strings: The model is exactly solvable if R depends on a parameter λ ∈ C such that eq.(2.2) is satisfied. Infact, if this is the case, one can introduce an infinite set of commuting operators as follows. First define the monodromy matrix: that can be viewed as an n×n-matrix with values in End(V N ) whose trace tr(T (λ)) ∈ End(V N ) is the transfer matrix (the partition function, we recall, is the trace in V N of the transfer matrix). The starting point of the QISM is the relation which follows from the definition of T and the QYBE. This relation implies the commutativity of the transfer matrices hence the existence of an infinite number of conserved quantities, typically regarded as the expansion coefficients lnT (λ) = ∞ n=1 C n λ n .
For quantum field theories in 1 + 1 dimension the argument is essentially the same with the change that one of the dimensions is time. This leads to consider on a different footing the two spaces in which the bonds take values instead of having two copies of the same space V (an n-dimensional space) both "horizontally" and "vertically" along the lattice. For instance one may think of the vertical space as the space of operators describing the time evolution of the quantum mechanical system (possibly an infinite-dimensional space); the N horizontal points represent the discretization of space and the continuum limit N → ∞ yields integrable quantum field theories. If we denote by V the vertical space to distinguish it from V , the horizontal space, then R ∈ End(V ⊗ V) is a Lax operator L and the QYBE reads where R 12 is a standard (i.e. numerical) matrix. If such a relation holds at each lattice point (i) and the "ultralocality" condition is satisfied then one can pursue the program as above: construct the monodromy matrix by multiplying the Lax operators and from it the commuting transfer matrices which lead to integrability.
We now introduce formally quasi-triangular Hopf algebras (quantum groups [1,15,16,17]). The property of quasi-triangularity, in particular, involves the definition of the universal R-matrix satisfying the Quantum Yang-Baxter equation without spectral parameters. The process of dressing such an R-matrix with parameters is called Yang-Baxterization, [18,19], and is not treated in the present paper.
Hopf Algebras involve both an algebric and a coalgebric structure. Let us start by showing the algebric structure in the following definition.
Definition 2.1. If G is a complex Lie algebra, then the extended enveloping algebra U q (G) of the universal enveloping algebra U (G) is the associative algebra over C with is the scalar product of the roots normalized or that for the short simple roots α we have (α, α) Comment. For a deeper analysis of the concepts involved in Definition 2.1 and in particular for an exhaustive discussion on the ring of functions of the Cartan generators necessary in the construction of the extended enveloping algebra we refer to Truini and Varadarajan [25].
Remark 2.3. The above construction works also when G is an affine Kac-Moody algebra (see Drinfeld [4]).
Convention. The subscript q in [m] q will be omited if no confusion can arise.
Comments. The extended enveloping algebra U q (G) is the "q-deformation" of the algebra U (G). Definition 2.1. may be used also for real forms (introducing the appropriate * -involutions) namely, where q ∈ R for the real compact forms (via the Weyl unitary trick) -e.g., for the classical compact algebras su(n), so(n), sp(n) -while for |q| = 1, it may be used if G is replaced by its maximally split forme.g., for the classical complex Lie algebras these forms being sl(n, R), so(n, n), so(n + 1, n), sp(n, R). (for the sense in which the limit is to be understood see Drinfeld [4]).
We recall that the Serre relations allow to define the universal enveloping algebra of a Lie algebra using only the simple roots and the Cartan matrix. It is a classical result that one thus gets an equivalent structure to the one obtained by taking the quotient of the tensor algebra with generators corresponding to all the roots by the full set of commutation relations (see Varadarajan [26]). There are two cases in which the explicit introduction of the generators associated to non-simple roots is convenient. One is the "q-analogue" of the Poincaré-Birkhoff-Witt theorem (which selects a representative element in each equivalence class of the quotient). The second is the universal R-matrix whose representations are related through Yang-Baxterization to the "physical R-matrix" which motivated the whole subject. Let us thus spend a few words on the generators associated to non-simple roots.
Conventions and notations. The Cartan subalgebra, spaned by H i , will be denoted by H, while the subalgebras spaned by X ± i will be denoted by G ± . We have the standard decompositions where Γ = Γ + ∪ Γ − is the root system of G and Γ + , Γ − the sets of positive, negative roots, respectively. Let us recall that the H i 's correspond to the simple roots α i of G, and if β = i n i α i , then to β correspond H β = i n i H i . The elements of G which span G β will be denoted by X β . These Cartan-Weyl generators are normalized, so that we have [X β , X −β ] = [H β ] qβ for β ∈ Γ + , q β = q (β,β)/2 .
(2.6) Remark 2.5. Instead of H i , some authors prefer to use K ± i defined by and then (2.3) and (2.4) become Remark 2.6. One can use, following Rosso [27], instead of X ± i , the generators E i and F i defined as follows  The comultiplication, counit, and antipode are defined on the generators of U q (G) as follows: whereρ ∈ H corresponds to ρ = 1 2 α∈Γ + α , Γ + being the set of positive roots, ρ = 1 2 α∈Γ + H α .
Comment. As it was remarked, for G = sl(2, C), in the paper [28] by Sklyanin, and, in general, in the papers [4,15] by Drinfeld and [16,17] by Jimbo, the algebra U q (G) is a Hopf algebra.
Remark 2.10. The opposite comultiplication and antipode from Remark 2.8. define In terms of the generators K ± i , E i , F i from Remarks 2.5. and 2.6., the relations One can also rewrite Serre's relations (2.5) as follows and L (resp. R) is the left (resp. right) multiplication.
Remark 2.11. ad q (E i ) acts as a twisted derivation.
and obeys also the relations where the indices indicate the embeddings of R into A ⊗ A ⊗ A is called a quasitriangular Hopf algebra.
Definition 2.6. From (2.9a) and one of (2.9b,c) it follows which is the Yang-Baxter equation for R without spectral parameters.
Comments. The universal R-matrix was given explicitly for G = sl(2, C) by Drinfeld [15], namely, This R-matrix is not in U q (sl(2, C)) ⊗ U q (sl(2, C)), since it contains power series involving X ± , but it is in some completion of it (in the h-adic topology used by Drinfeld [4,15] (q = e h )). This fact is valid for the R-matrices of all U q (G). Hopf algebras with such an R-matrix are called by Drinfeld [4] pseudo quasi-triangular Hopf algebras, and by Majid [7] essentially quasi-triangular Hopf algebras. For G = sl(n, C), an explicit formula for R was given by Rosso [27].

SUBSEQUENT AND RECENT DEVELOPMENTS
At the end of eighties other approaches to quantum groups were given. The objects of these approaches -which can be called quantum matrix groups -are Hopf algebras in duality to quantum algebras. such that, for a, b ∈ A and a ′ , b ′ ∈ A ′ the following relations hold One of these approaches is due to Faddeev, Reshetikhin and Takhtajan [29,30] and it is called "R-matrix approach". It is based on the main relation of the quantum inverse scattering method. The quantum group matrices play the role of the quantum monodromy matrices (with operator-value entries) of the auxiliary linear problem and the Yang-Baxter equation is the compatibility equation.
Another approach is that of Manin [31,32,33], who considers quantum groups as symmetries of non-commutative, or quantum, spaces. The resulting objects are the same as those of the first approach.
A third approach that we would like to mention here is that of Woronowicz [34,35,36]. This approach also deals with same objects with some additional structures since Woronowicz's starting point is the theory of C * -algebras.
Note. Connections between the above mentioned approaches can be found in the papers by Doebner, Hening and Lücke [37], Majid [7], and Rosso [22].
At the end of eighties, Drinfeld [15], inspired by Knizhnik-Zamolodchikov equations (see [38]), developed a theory of formal deformations and introduced a new notion of "quasi-Hopf algebras".
The matrix quantum group approaches were recently developed, in particular finding consistent multiparametric deformations. This is related to the development by Wess, Zumino and collaborators of differential calculus on quantum hyperplanes.
The latter approach is actually an example of non-commutative differential geometry which in opinion of Manin [39] is different in spirit from that of Connes [40], although the exact relation is not known yet. For a detailed review of the above mentioned developments we refer the reader to § §2-5 from Dobrev's paper [3].
Finally, we like to mention the very recent paper by Mack and Schomerus [41] on "quantum field planes". These objects are generalizations of the quantum planes which were studied by Manin, Wess and Zumino, and others, and were generalized to the quasi-associative case by the same authors [42]. Basically, the construction of quantum field planes replaces the ground field C of quantum planes by the noncommutative algebra of observables of the quantum field theory in the local Lorentz frame.
Comments. An interesting open problem could be to reconsider Mack-Schomerus construction in case when the algebra of observables is a (non-commutative) Jordan algebra.
Remark 3.1. As a final note to this survey on the history of quantum groups we want to emphasize that quantum groups are involved and studied in many different fields of mathematics and physics some of which we haven't even mentioned so far. Among them are: topological quantum field theories, 2-dimensional gravity and 3-dimensional Chern-Simons theory [43,44,45,7], rational conformal field theory [46,47,48], braid and knot theory [49,50], non-standard quantum statistics [51], quantum Hall effect [52,53].

RELATIONS WITH JORDAN STRUCTURES
We would like to start by recalling here the new topic proposed by Truini and Varadarajan at the end of their recent paper [54], namely: quantization of Jordan structures.
Let us explain why it is interesting to investigate this new topic.
It is known that the quantization of the Poincaré group is receiving more and more attention by the physicists. The main reason of this attention consists in its relationship with the non-commutative geometry of quantum Minkowski space (see [55]). It is believed that the models of non-commutative space-time, and their quantum symmetry groups, may provide a basis for building a divergence free theory of elementary particles and their fields, including gravitation, thus overcoming the difficulties arising out of the structure of conventional space-time at small distances.
Consequently, Truini and Varadarajan (see [54, p.732 On the other hand, let us point out that Jordan pairs are much more common than what one would expect. So, there exists a 1-1 correspondence between 3-graded Lie and Jordan pairs V (plus the choice of L 0 between Der (V ) and Inder (V )) -see, for instance, McCrimmon [56, p.622]. As it is known, many Lie algebras are three-graded (e.g., A(2), C(3), E (7)).
In this respect, it is interesting to point out that Okubo [57] has related triple products, which are linearization of the quadratic maps, to the quantum R-matrix and used the relationship to find solutions to the quantum Yang-Baxter equation.
More recently, in a series of papers, Okubo [58,59,60,61] reformulated first the Yang-Baxter equation as a triple product relation and then solved it for triple systems called "orthogonal" and "symplectic" (see Okubo [58,59]). A superspace extension of this work was given by Okubo himself in [60,61].
In a very recent paper, Okubo [62] gave some solution of Yang-Baxter equation in terms of Jordan triple systems and of so-called "anti-Jordan triple systems". (see where δ = ±1. The case δ = 1 defines the well known (linear) Jordan triple systems (see, for instance, Meyberg [63]), while the case δ = −1 defines the anti-Jordan triple systems.
Note. As it was noticed by Koecher [64], a glimps of Jordan triple system was given Comments. Compare the definition of "anti-Jordan triple systems" given by Okubo [62] (see Definition 4.1.) with the definition of "anti-Jordan pairs" given by Faulkner and Ferrar in [65].
In view of these, the solution is likely related [66] to some exactly solvable twodimensional quantum field theory.
Finally we want to mention a result by Svinolupov [67] which is interesting in the context of this paper. He considers systems of nonlinear equations which, in a particular case, may be reduced to the nonlinear Schrödinger equation and are therefore called generalized Schrödinger equations. A one to one correspondence between such integrable systems and Jordan Pairs is established. It turns out that irreducible systems correspond to simple Jordan Pairs.
In our opinion the general setting in which one should consider the problem of quantizing the Jordan structures is that of Jordan Pairs. We are currently investigating this possibility with the belief that a quantum analog of the classical link between Jordan and Lie structures would give a deeper insight and reveal new aspects in the theory of quantum groups.