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Affine Kac-Moody symmetric spaces associated with untwisted Kac-Moody algebras

In this paper we have computed all the affine Kac-Moody symmetric spaces which are tame Frechet manifolds starting from the Vogan diagrams related to the affine untwisted Kac-Moody algebras. The detail computation of affine Kac-Moody symmetric spaces associated with A_1^(1) and A_2^(1) are shown algebraically to corroborate our method.


I. INTRODUCTION
Finite dimensional symmetric spaces [20,25] are rather well understood mathematical objects which have recently gained much importance in both mathematics and physics due to their intimate connections with random matrix theories, Reimannian geometries and their applications to many integrable systems, quantum transport phenomena (disordered system etc) [4,5,9,22]. A compact irreducible symmetric space is either a compact simple Lie group G or a quotient G/K of a compact simple Lie group by the fixed point set of an involution ρ (or an open subgroup of it) and g = t ⊕ p is the decomposition of the Lie algebra g of group G into +1 and -1 eigenvalue spaces of ρ then K acts on g by adjoint representation leaving the decomposition invariant. The restriction of this action to p can be identified with the isotropy representation of G/K and we know that the isotropy representation of a symmetric space is polar.
With the advent of Kac-Moody algebras [15,16,21] which can be considered as the generalization of finite dimensional Lie algebras, naturally a search of infinite dimensional version of symmetric spaces began, the closest generalization being the affine Kac-Moody symmetric spaces. An affine Kac-Moody symmetric space is by definition either an affine Kac-Moody groupĜ(group type) or a quotientĜ/Ĝρ ofĜ by the fixed point set of an involution of the second kind. In fact ifĝ =t +p in the splitting of the Lie algebra ofĜ into the ±1 eigen spaces ofρ. Then the metric ofĜ/K is the left invariant metric obtained from the restriction of the inner product ofĝ top. In finite dimension, the isotropy representation is polar itself while in infinite dimension it leaves invariant a co-dimension-2 sub-manifold which can be identified with a (pre-)Hilbert space and the induced action on this space is a polar action by affine isometries.
The study on affine Kac-Moody symmetric spaces began with C.L.Terng [26] who conjectured the existence of infinite dimensional symmetric spaces. Important progresses towards their constructions and geometries are achieved by B. Popescu [23], W. Freyn [10], Heintze [13,14], et al. Now it has been shown that affine Kac-Moody symmetric spaces are tame Fréchet manifolds. In particular let G be the simply connected Lie group with Lie algebra g and denote σ be the automorphism of G corresponding to σ ∈ Aut(g). Then the loop group L(G, σ) = {g : R → G | g ∈ C ∞ , g(t + 2π) = σ(g(t)) ∀t} (1) with point wise multiplication is a Fréchet Lie group with Lie algebra L(g, σ). The affine Kac-Moody groupL(G, σ) will be a T 2 = S 1 × S 1 bundle over L(G, σ). By constructionL(G, σ) is a Fréchet group and it is a Lorentz manifold. It is also well known thatL(G, σ)/L(G, σ)ρ for any involutionρ ofL(G, σ) and L(G, σ)/L(G, σ) ρ for any involution ρ are tame Fréchet. The classification of affine Kac-Moody symmetric spaces is essentially equivalent to the classification of involutions of affine Kac-Moody algebras upto conjugation. This has been achieved by a long series of papers by various authors Batra [1], Levstein [19], Kobayashi [17], Rousseau and Messaoud [24]. There is a one-to-one correspondence between real forms of Kac-Moody algebras, involutions. The diagramatic representation of real forms are the Vogan diagrams (to each real form there is a unique Vogan diagram) which are Dynkin diagrams together with some extra pieces of informations. We have collected/construted the Vogan diagrams related with untwisted classical algebras A (1) n , B (1) n , C (1) n , D (1) n explicitly. From the Vogan diagrams we have obtained the fixed point set as well as the real forms of the algebras and then we have constructed affine Kac-Moody symmetric spaces.
In chapter-2 we have given a brief introduction to affine untwisted Kac-Moody algebras with their involutions, real forms and Vogan diagrams.
In chapter-3 we have explicitly calculated the affine Kac-Moody symmetric spaces with two elementary untwisted Kac-Moody algebras A (1) 1 and A (1) 2 relating them with their Vogan diagrams to corroborate our technique to construct * Electronic address: kcpati@nitrkl.ac.in affine Kac-Moody symmetric spaces. Towards the end of the chapter we have given exhaustive list of all the affine Kac-Moody symmetric spaces together with their real forms and fixed algebras. Chapter-4 contains appendix.

A. Kac-Moody Lie algebras
Let I = [1, n + 1], n ∈ N, be an interval in N. A matrix A = (a ij ) i,j∈I with integer coefficients is called a generalized Cartan matrix if it satisfies the following conditions: 1. a ii = 2 for i = 1, · · · , n + 1.
A realization of A is a triple (h, π,π), where h is a finite-dimensional complex vector space and π = {α i } i∈I ⊂ h * anď π = {α i } i∈I ⊂ h are indexed subsets in h * and h respectively, and they satisfy 1. both π,π are linearly independent.

rank (A) = 2n -dim h.
For any n × n matrix A there exits a unique(upto isomorphism) realization.
Given two matrices A and A ′ and their realizations (h, π,π) and (h ′ , π ′ ,π ′ ), we obtain a realization of the direct sum of two matrices A matrix A is called decomposable if after reordering of indices A decomposes into a non-trivial direct sum. Otherwise A is called indecomposable.
Let A = (a ij ) be a generalized Cartan matrix and let (h, π,π) be a realization of A. Let g(A) be a complex Lie algebra with generators e i , f i for i = 1 · · · n and h and the following defining relations: The Lie algebra g = g(A) is called a Kac-Moody algebra. The subalgebra h of g is called the Cartan subalgebra. The matrix A is the Cartan matrix of g which is of rank n. The elements e i , f i for i = 1 · · · n, are called Chevalley generators and they generates the subalgebra g ′ = [g, g] and g = g ′ + h.

B. Affine Kac-Moody Algebras
Consider the generalized Cartan matrix A. It is called a Cartan matrix of affine type if • A is an indecomposable matrix, i.e. after the indices are reordered A cannot be written in the form • There exits a vector (a i ) n+1 i=1 , with a i all positive such that A(a i ) n+1 i=1 = 0. Then, the algebra g associated with A is called an affine Kac-Moody algebra. Affine Kac-Moody algebras are of two types untwisted and twisted. In this paper we have confined ourselves to untwisted case only.

C. A realization of non-twisted affine Kac-Moody Lie algebra
Let L = C[t, t −1 ] be the algebra of Laurent polynomials in t. The residue of the Laurent polynomial P = j∈Z c j t j (where all but finite number of c j are zero) is ResP = c −1 . Letg be a finite-dimensional simple Lie algebra over C of type X n , then L(g) = L ⊗g is an infinite-dimensional Lie algebra with the bracket Fix a non-degenerate, invariant, symmetric bilinear form (., .) ing and extend this form to an L valued form (., .) t on L(g) by The derivation t j (d/dt) of L extends to L(g) by Therefore ψ(a, b) = Res( da dt , b) t for a, b ∈ L(g) defines a two-cocycle on L(g). Now we denote byL(g) the central extension of the Lie algebra L(g) associated to the cocycle ψ. Explicitlỹ Finally, denote byL(g) is the Lie algebra which is obtained by adjoining toL(g) a derivation d which acts on L(g) as t d dt and kills c. Explicitly we haveL(g) = L(g) ⊕ Cc ⊕ Cd with the bracket defined by where x, y ∈g; λ, µ, λ 1 , µ 1 ∈ C; j, k ∈ Z. ThisL(g) is a non-twisted affine Kac-Moody Lie algebra associated to the affine matrix A of type X n .

D. Automorphisms and Real forms of non-twisted affine Kac-Moody algebras
Define a group G acting on the algebra g through adjoint representation Ad : G → Aut(g). It is generated by the subgroup U α for α ∈ ±π and Ad(U α ) = exp(ad(g α )).
A maximal ad g -diagonalizable subalgebra of g is called a Cartan subalgebra. Every Cartan subalgebra of g is Ad(G)-conjugate to the standard Cartan subalgebra h. A Borel subalgebra of g is maximal completely solvable subalgebra. It is conjugated by Ad(G) to b + and b − where b + = h ⊕ α>0 g α and b − = h ⊕ α<0 g α . However b + and b − are not conjugated under Ad(G). So there are two conjugacy classes of Borel subalgebra the positive and negative subalgebras.
A real form of g is a algebra g R over R such that there exists an isomorphism from g to g R ⊗ C. If we replace C by R in definition of g then we obtain a real form of g R which is called split real form.
An automorphism σ of g is called an involution if σ 2 = Id. The involution is called semi-linear if σ(λx) =λσ(x) for λ ∈ C and x ∈ g. A real form of g correspondences to a semi-linear involution of g. A linear or semi-linear automorphism σ of g is said to be of first kind if σ(b + ) is Ad(G)-conjugate to b + and it is of second kind if σ(b + ) is Ad(G)-conjugate to b − . Any automorphism of g is either an automorphism of first kind (type 1) or an automorphism of second kind (type 2).
Let g R be a real form of g. Fix an isomorphism from g to g R ⊗ C. Then the Galois group Γ = Gal(C/R) acts on g and the corresponding group G. Then g R can be identified with the fixed point set g Γ . If Γ consists of first kind automorphism then we say g R is almost split, otherwise if the non-trivial element of Γ is of second kind automorphism then we say g R is almost compact(non-compact). Denote the group of C-linear or semilinear automorphisms of g as Aut R (g). The group Aut(g) is normal in Aut R (g) and of index 2. A semilinear automorphism of order 2 of g is called a semiinvolution of g.
Definition II.1. Let σ ′ be a semi-involution of g of second kind and let g R = g σ ′ be the corresponding almost compact real form. A Cartan semi-involution ω ′ which commutes with σ ′ is called a Cartan semi-involution for σ ′ or g R . The involution σ = σ ′ ω ′ is called a Cartan involution of σ ′ and also the restriction ω R ′ of σ to g R is called Cartan involution for g R .
The algebra of fixed points t 0 = g σ R is called a maximal compact subalgebra of g R . Now we have the Cartan decomposition g R = t 0 ⊕ p 0 and t 1 = t 0 ⊕ ip 0 where p 0 is the eigenspace of ω ′ R for eigen value −1. Let t 0 be a maximal abelian subspace of t 0 . Then h 0 = Z g R (t 0 ) is a σ-stable Cartan subalgebra of the almost compact real form g R of the form h 0 = t 0 ⊕ a 0 with a 0 ⊆ p 0 .
Definition II.2. A σ-stable Cartan subalgebra h 0 = t 0 ⊕ a 0 with t 0 ⊆ t 0 and a 0 ⊆ p 0 of an almost compact real form g R is maximally compact if the dimension of t 0 is as large as possible and it is maximally non-compact if the dimension of a 0 is as large as possible.
A maximally compact Cartan subalgebra h 0 of an almost compact real form g R has the property that all the roots are real on a 0 and imaginary on t 0 . One says that a root is real if it takes real value on h 0 = t 0 ⊕ a 0 , i.e. vanishes on t 0 . It is imaginary if it takes imaginary value on h 0 , i.e. vanishes on a 0 and complex otherwise.

E. Classification of Real forms
Under Aut(g) there is a one-one correspondence between the conjugacy classes of involutions(linear) of second kind of g and the conjugacy classes of almost split real forms of g. Again there is a bijection between the conjugacy classes under Aut(g) of semi-involution of second kind and conjugacy classes of involution of first kind. Thus one obtain under Aut(g) a one-to-one correspondence between conjugacy classes of linear involutions of first kind(including identity) and the the conjugacy classes of almost compact real forms of g. The compact real form is unique and corresponds to the identity.
Let h 0 be a σ-stable Cartan subalgebra of g R .Then there are no real roots iff h 0 is maximally compact. Let g R be almost compact real form of g corresponding to the semi-involution of the second kind σ ′ of g. Let σ be the Cartan involution of g R and let g R = t 0 ⊕ p 0 be the corresponding Cartan decomposition [3,12,28]. Let t 0 be maximal abelian subspace of t 0 . Then h 0 = Z g R (t 0 ) is a σ-stable Cartan subalgebra of g R of the form h 0 = t 0 ⊕ a 0 with a 0 ⊆ p 0 . This h 0 is a maximally Cartan subalgebra of g R because t 0 is as large as possible.
Theorem II.3. (Theorem 45 [10]) Let g be an complex affine Kac-Moody algebra and C be a real form of it which is compact type. The conjugacy classes of real forms of non compact type of g are in bijection with the conjugacy classes of involutions on C. The correspondence is given by C = K ⊕ P → K ⊕ iP where K and P are the ±-eigen spaces for the involution.
However every real form is either of compact type or of non-compact type, a mixed type is not possible.
Lemma II.4. (Lemma 47 [10]) Let g R be a real form of non-compact type. Let g R = K ⊕ P be Cartan decomposition. The Cartan Killing form is negative definite on K and positive definite on P .

F. Vogan diagrams
For classification of real forms of affine Kac-Moody algebra there are two main approaches: One focuses the maximal non-compact Cartan subalgebra that leads to Satake diagrams [27]. The other one is on maximal compact Cartan subalgebra that leads to Vogan diagrams [2,6,18].
Let almost compact real form g R of g and σ be the Cartan involution on g R leading to the Cartan decomposition g R = t 0 ⊕ p 0 . Let h 0 be the maximally compact σ-stable Cartan subalgebra of g R with complexification h = t ⊕ a. Let us denote ∆ = ∆(g, h) be the set of roots of g with respect to h. This set doesn't contain any real root as h 0 is assumed to be maximally compact. From ∆ we choose a positive system ∆ + that takes it 0 before a. since σ is +1 on t 0 and −1 on a 0 and since there are no real roots σ(∆ + ) = ∆ + . Therefore σ permutes the simple roots. It fixes the simple roots that are imaginary and permutes in 2-cycles the simple roots that are complex.
Definition II.5. By Vogan diagram of the triple (g R , h 0 , ∆ + ) we mean the Dynkin diagram of ∆ + with the 2element orbits under σ labelled an arrow and with the 1-element orbit painted or not depending upon whether the corresponding imaginary simple root is non-compact or compact.
Every Vogan diagram represents an almost compact(non-compact) real form of some affine Kac-Moody Lie algebra. Two diagrams may represent isomorphic algebras and in that case the diagrams are equivalent. So the classification of Vogan diagram gives rise to the classification of almost compact real form of affine Kac-Moody Lie algebra.
The equivalence of Vogan diagram is defined as the equivalence relation generated by the following two operations: 1. Applications of an automorphism of the Dynkin diagram.
2. Change in the positive system by reflection in a simple, non-compact root, i.e. by a vertex which is colored in the Vogan diagram.
As a consequence of reflection by a simple non-compact root α, the rules for single and triple lines is that we have α colored and its immediate neighbour is changed to the opposite color. The rule for double line is that if α is the smaller root, then there is no change in the color of immediate neighbour, but we leave α colored. If α is a bigger root, then we leave α colored and the immediate neighbour is changed to the opposite color. If two Vogan diagrams aren't equivalent to each other, then they are called non-equivalent.
Definition II.6. An abstract Vogan diagram is an irreducible abstract Dynkin diagram of non-twisted affine Kac-Moody Lie algebra with two additional piece of structure as follows: 1. One is an automorphism of order 1 or 2 of the diagram, which is indicated by labelling the 2-element orbits.
2. Second one is a subset of 1-element orbits which is to be indicated by pointing the vertices corresponding to the members of the subset.
Every Vogan diagram is an abstract Vogan diagram. It is always convenient to represent equivalence class of Vogan diagrams with minimum number of vertices painted. We have Borel Seibenthal theorem for affine Kac-Moody algebras [7] which states that Theorem II.7. Every equivalence class of Vogan diagram has a representative with atmost two vertices painted.

Some more important results:
Theorem II.8. If an abstract Vogan diagram for an non-twisted affine Kac-Moody Lie algebra is given, then there exits an almost compact real form of a non-twisted affine Kac-Moody Lie algebra such that the given diagram is the Vogan diagram of this almost compact real form.
Theorem II.9. If two almost compact real forms of a non-twisted affine Kac-Moody Lie algebra g have equivalent Vogan diagram then they are isomorphic.

III. AFFINE KAC-MOODY SYMMETRIC SPACE
In this chapter we briefly review with the definition and geometry of the affine Kac-Moody symmetric spaces [10,13,23] with explicit determination of affine Kac-Moody symmetric spaces associated with A Definition III.1. A tame Fréchet manifold M with a weak metric having a Levi-civita connection is called a symmettric space, iff ∀p ∈ M there is an involution isometry ρ p , such that p is an isolated fixed point of ρ p .  We can summarize all the results we have discussed so far as follows: There is an one-one correspondence between the conjugacy classes of involution of second kind of affine Kac-Moody algebra g and almost split real form and also between the conjugacy classes of involution of first kind and almost compact real form of g. Again to each real form there is a unique Vogan diagram. On the other hand all real forms are of two types: compact, non-compact and the conjugacy classes of real forms of non-compact type of g are in bijection with the conjugacy classes of involution on the compact real form (C). So the affine Kac-Moody symmetric spaces can be classified using compact real form and involution of second kind. Now we can conclude that classification affine Kac-Moody symmetric spaces are intimately linked with classification of Vogan diagram. Starting with the compact real form C we can construct all the various non-compact real forms by applying involutive automorphisms to C followed by Weyl unitary trick.
A. Affine Kac-Moody symmetric spaces associated with A The chevelley generators for A (1) 1 are given by:

The Cartan involution of
As a result the compact form is generated by: Explicitly the compact real form is given by this is a skew hermitian matrix with trace zero, which is identified as su (1) (2). The Vogan diagram associated with this real form is given by the following: Now the general form [8] of an involutive automorphism associated with a affine Kac-Moody algebra is given as: for type 1(a) automorphism and for type 1(b) automorphism But the conjugacy class of type 1(b) automorphisms with u = 1 and u = −1 correspond to some automorphisms of type 1(a) with u = 1 and u = −1 respectively. Also we shall like to mention that type 2(a) and type 2(b) automorphisms are obtained by composing type 1(a) and 1(b) with Cartan involution respectively. Action of σ on c is σ(c) = µc, however for 1(a) automorphism µ = 1. Now and for our cases σ(d) = d except the case-III of A Case-I: If then under the automorphism (8) with (10) a general matrix with block matrices A, B, C, D transforms as Hence the fixed subalgebra K of C(t) is given by Hence the non- 1 (1, 1) and the corresponding Vogan diagram is given by Case II: Similarly if then under the automorphism(8) with (12) the matrix A B C D trasforms as We observe when n is an even integer it reduces to case I, giving same real form. But n is odd then we have .
Hence the non- −1 (1, 1). The corresponding Vogan diagram is the given by The affine Kac-Moody symmetric spaces are: Case III: Now consider So now under the automorphism (8) with (14) a general matrix A B C D transform as, Then from a simple mathematical manipulation we observe that in this case Now putting the values of A, B, C and D in K we get with K satisfies K * + K = 0 and hence K is identified as so (1) (2). So we have K + iP = Hence the affine Kac-Moody symmetric spaces are B. Affine Kac Moody symetric space associated with A is the following: Hence the Compact form is generated by which is a skew hermitian matrix with trace zero it is identified as su (1) (3). The Vogan diagram is Now proceeding similarly as A 1 case and taking different cases we have, Case I: Under the automorphism (8) with (19) the matrix transforms as Here The decomposition of K as in (??) shows that is isomorphic to su with Z 2 = iB is a 2 × 2 matrix, Z 1 = A is a 2 × 2 skew hermitian matrix and Z 3 = C is a 1 × 1 skew hermitian matrix and also satisfies T rZ 1 + T rZ 3 = 0. Hence the non-compact real form is K + iP ⊕ Ric ⊕ Rid ∈ su (1) Thus the corresponding symmetric spaces are: Case II: Under the automorphism (8)

with (24) the matrix transforms as
Here Thus K + iP is with Z 2 = iB is a 2 × 1 matrix, Z 1 = A is a 2 × 2 skew hermitian matrix and Z 3 = C is a 1 × 1 skew hermitian matrix and also satisfies T rZ 1 + T rZ 3 = 0. Hence the non-compact real form is K + iP ⊕ Ric ⊕ Rid ∈ su Thus the corresponding symmetric spaces are: Case III: Under the automorphism (8) such that trace of this matrix is zero. Hence the non-compact real form is Thus the corresponding symmetric spaces are Case IV: Under the automorphism (8) with (33) the matrix transforms as such that trace of this matrix is zero. Hence the non-compact real form is K + iP ⊕ Ric ⊕ Rid ∈ sl Thus we have completed the explicit (algebraically) calculation of affine Kac-Moody symmetric spaces associated with A 1 and A 2 and relating them with the Vogan diagrams which classify their real forms also. The next tables contain the real forms, Vogan diagrams, fixed algebras associated with the automorphism, related compact and noncompact affine kac-Moody symmetric spaces. In order to have a clear idea about the computation of fixed algebras and affine Kac-Moody symmetric spaces we have provided an appendix which contains all the necessary ingredients to understand the method of our calculations.
so (1) σv (n+1,n−1) αn P P P P P P P P P αn P P P P P P P P P α 1 P P P P P P P P P so (1) σs (1, 2n−1) In this subsection we have given the irreducible reduced root system of complex semi simple Lie algebras A n for n ≥ 1, B n for n ≥ 2, C n for n ≥ 3 and D n for n ≥ 4. In case of A n the under lying vector space V = {v ∈ R n+1 |< v, e 1 + · · · + e n+1 >= 0} and for rest algebras V = R n .∆ denotes the root system which is a subspace of some R k = k i=1 a i e i . Here {e i } is the standard orthonormal basis and a i 's are real. ∆ + is the positive root system and Π is simple root system. The diagram automorphism for affine Lie algebra are as follows. For A (1) n , the automorphism group of the Dynkin diagram is the dihedral group D n+1 which is generated by reflection s : i −→ n + 1 − i(mod n + 1) and the rotation r : i −→ i + 1(mod n + 1) which is of order r + 1. For D (1) r , the automorphism group is generated by the vector automorphism σ v , the spinor automorphism σ s and the conjugation γ. σ v acts as 0 ←→ 1, r ←→ r − 1 and i −→ i else, and hence is of order 2. The map γ acts as r ←→ r − 1 and i −→ i else. If r is even, σ s acts as i −→ r − i is of order 2 while for odd r the prescription i −→ r − i only holds for 2 ≤ i ≤ r − 2 and is supplemented by 0 −→ r −→ 1 −→ r − 1 −→ 0. For the untwisted algebra g = B r , there is only a single non-trivial automorphism γ which is a reflection [11].

C. Classical Non-compact real affine Kac-Moody Lie Algebras
Let g R be a real affine Kac-Moody Lie algebra and t 0 be the fixed subalgebra of a Cartan involution on g R . c 0 is the center of t 0 , then simple roots of t 0 are obtained as follows. When the automorphism in the Vogan diagram is non trivial, we know that t 0 is semisimple. The simple roots for t 0 then include the compact imaginary simple roots and the average of the members of each 2-element orbit of simple roots. If the Vogan diagram has no painted imaginary root, there is no other simple root for t 0 . Otherwise there is one other simple root for t 0 , obtained by taking a minimal complex root containing the painted imaginary root in its expansion and averaging it over its 2-element orbit under the automorphism. When the automorphism is trivial, either dim c 0 = 1, in this case the simple roots for t 0 are the compact simple roots for g 0 , or else dim c 0 = 0, in this case the simple roots for t 0 are the compact imaginary simple roots for g 0 and one other compact imaginary root. In latter case this other compact imaginary root is the unique smallest root containing the non-compact simple root twice in its expansion. We have discussed below in detail, for each real algebra separately.