Symmetrizable Kac Moody Algebra Research Articles

Tensor product stabilization in Kac–Moody algebras

We consider a large class of series of symmetrizable Kac–Moody algebras (generically denoted X n ). This includes the classical series A n as well as others like E n whose members are of Indefinite type. The focus is to analyze the behavior of representations in the limit n → ∞ . Motivated by the classical theory of A n = sl n + 1 C , we consider tensor product decompositions of irreducible highest weight representations of X n and study how these vary with n . The notion of “double-headed” dominant weights is introduced. For such weights, we show that tensor product decompositions in X n do stabilize, generalizing the classical results for A n . The main tool used is Littelmann's celebrated path model. One can also use the stable multiplicities as structure constants to define a multiplication operation on a suitable space. We define this so-called stable representation ring and show that the multiplication operation is associative.

ON THE NILPOTENCY OF CERTAIN SUBALGEBRAS OF KAC-MOODY ALGEBRAS OF TYPE AN(r)

Let (equation omitted) be a symmetrizable Kac-Moody algebra with the indecomposable generalized Cartan matrix A and W be its Weyl group. Let be the highest root of the corresponding finite dimensional simple Lie algebra of g. For the type , we give an element such that . And then we prove that the degree of nilpotency of the subalgebra (equation omitted) is greater than or equal to .

A sufficient condition for pairing problem of generators in symmetrizable Kac–Moody algebras

In a symmetrizable Kac–Moody algebra g ( A ), let α =∑ i =1 n k i α i be an imaginary root satisfying k i >0 and 〈 α , α i ∨ 〉<0 for i =1,2,…, n . In this paper, it is proved that for any x α ∈ g α ⧹{0}, satisfying [ x α , f n ]≠0 and [ x α , f i ]=0 for i =1,2,…, n −1, there exists a vector y such that the subalgebra generated by x α and y contains g ′( A ), the derived subalgebra of g ( A ).

CERTAIN MODULES WITH TWINING MAPS AND DECOMPOSITION RULES OF LITTELMANN TYPE

We consider certain modules, for which twining characters can be defined, over a symmetrizable Kac-Moody algebra , and show a kind of complete reducibility theorem for them. We also prove two kinds of decomposition rules of Littelmann type for twining characters by using the “path model” for the twining character of an integrable highest weight -module.

Combinatorial Models for Weyl Characters

We present a set of axioms for combinatorial objects closely related to those for Kashiwara's crystals. We show that any model for the axioms, such as Littelmann's path model, has a character—a nonnegative sum of irreducible characters for a semisimple Lie group or algebra, or more generally, a symmetrizable Kac–Moody algebra. Moreover, there are simple explicit restriction rules and rules for decomposing the product of any such character by an irreducible character.

Lakshmibai–Seshadri Paths Fixed by a Diagram Automorphism

We prove that the Lakshmibai–Seshadri paths fixed by a diagram automorphism ω of a Kac–Moody algebra can be identified with the Lakshmibai–Seshadri paths for the orbit Lie algebra, by introducing ω-root operators. We also give certain formulas for the twining characters of integrable highest weight modules over a symmetrizable Kac–Moody algebra and of Demazure modules for a finite-dimensional semi-simple Lie algebra.

Polytopes Associated to Demazure Modules of Symmetrizable Kac–Moody Algebras of Rank Two

Let ω1,ω2 be the two fundamental weights of a symmetrizable Kac–Moody algebra g of rank two (hence necessarily affine or finite), and τ an element of the Weyl group. In this paper we construct polytopes Pτ(ω1),Pτ(ω2)⊂Rl(τ) and a linear map ξ: Rl(τ)→h* such that for any dominant weight λ=k1ω1+k2ω2, we have CharEτ(λ)=eλ∑eξ(x), where the sum is over all the integral points x, of the polytope k1Pτ(ω1)+k2Pτ(ω2). Furthermore, we show that there exists a flat deformation of the Schubert variety Sτ into the toric variety defined by Pτ(ω1),Pτ(ω2).

Universal Verma modules andW-resolvents over Kač-Moody algebras

We consider new aspects of extremal equations over symmetrizable Kac-Moody algebras. We develop new methods (reproducing classical finite-dimensional results) that can be applied to infinite-dimensional (affine) Lie algebras. We describe special extensions of universal enveloping algebras, investigate the fine structure of W-resolvents, and use these methods to investigate “extremal projectors” over Kac-Moody algebras.

Twisted traces of intertwiners for Kac-Moody algebras and classical dynamical r-matrices corresponding to generalized Belavin-Drinfeld triples

In early eighties, Belavin and Drinfeld showed that nonskewsymmetric classical r-matrices for simple Lie algebras are classified by combinatorial objects which are now called Belavin-Drinfeld triples. Later the second author of the present paper generalized this result to the case of dynamical r-matrices, and showed that they correspond to generalized Belavin-Drinfeld triples. The dynamical r-matrix corresponding a triple is given by a certain explicit formula, which works for any symmetrizable Kac-Moody algebra. In special cases, this formula gives the Felder and the Belavin r-matrix. In 1994, Felder associated with every dynamical r-matrix a system of differential equations called the KZB equations. In the case of the Felder and Belavin r-matrix, solutions of these equations are known to have a representation theoretical presentation, as conformal blocks (i.e. traces of products of intertwining operators between representations of Lie algebras). In this paper, we propose such a presentation for ANY dynamical r-matrix given by the second author's formula. In particular, the paper sheds light on the Belavin-Drinfeld classical r-matrices of 1984. This paper begins a series of papers. In the next paper, joint with T.Schedler, we will give an explicit quantization of the dynamical r-matrices for simple Lie algebras (in particular, Belavin-Drinfeld r-matrices), which has been an open problem for a number of years. In another paper, we plan to generalize the results of the present paper to quantum groups. This should yield quantum KZB equations which involve quantum R-matrices obtained by quantizing the classical r-matrices that appear in this paper.

Sur une formule des caractères pour les algèbres de Kac-Moody symétrisables

Let g be a symmetrizable Kac-Moody algebra. In [KW1], Kac and Wakimoto proved a character fomula for a large class of irreducible highest weight g -modules which includes integrable modules as a special case. We obtain an extension of Kac and Wakimoto's formula.

Résultats combinatoires sur les modules de Demazure

In this paper, we introduce the notion of polytopes with integral structure, which generalizes the notion of usual polytopes whose vertices have integral coordinates. Then we consider a symmetrizable Kac-Moody algebra. For every dominant weight λ and every element of the Weyl group w, we construct a polytope with integral structure such that its Ehrhart polynomial describes the dimension of the Demazure module of highest weight nλ associated to w.

From Dynkin diagram symmetries to fixed point structures

Any automorphism of the Dynkin diagram of a symmetrizable Kac-Moody algebra g induces an automorphism of g and a mappingτ ω between highest weight modules of g. For a large class of such Dynkin diagram automorphisms, we can describe various aspects of these maps in terms of another Kac-Moody algebra, the “orbit Lie algebra” ğ. In particular, the generating function for the trace ofτ ω over weight spaces, which we call the “twining character” of g (with respect to the automorphism), is equal to a character of ğ. The orbit Lie algebras of untwisted affine Lie algebras turn out to be closely related to the fixed point theories that have been introduced in conformal field theory. Orbit Lie algebras and twining characters constitute a crucial step towards solving the fixed point resolution problem in conformal field theory.

Quantum affine algebras and affine Hecke algebras

One of the most beautiful results from the classical period of the representation theory of Lie groups is the correspondence, due to Frobenius and Schur, between the representations of symmetric groups and those of general or special linear groups. If V0 is the natural irreducible (n+ 1)–dimensional representation of SLn+1(CI), the symmetric group Sl acts on V ⊗l 0 by permuting the factors. This action obviously commutes with the action of SLn+1(CI). It follows that one may associate to any right Sl–module M a representation of SLn+1(CI), namely FS(M) = M⊗SlV ⊗l 0 , the action of SLn+1(CI) on FS(M) being induced by its natural action on V ⊗l 0 . The main result of the Frobenius–Schur theory is that, if l ≤ n, the assignment M → FS(M) defines an equivalence from the category of finite–dimensional representations of Sl to the category of finite–dimensional representations of SLn+1(CI), all of whose irreducible components occur in V ⊗l 0 . Around 1985, Drinfeld and Jimbo independently introduced a family of Hopf algebras Uq(g), depending on a parameter q ∈ CI , associated to any symmetrizable Kac–Moody algebra g. Assuming that q is not a root of unity, Jimbo [7] proved an analogue of the Frobenius–Schur correspondence in which SLn+1(CI) is replaced by Uq(sln+1), V0 by the natural (n+1)–dimensional irreducible representation V of Uq(sln+1), and Sl by its Hecke algebra Hl(q 2). In [5], Drinfeld announced an analogue of the Frobenius–Schur theory for the Yangian Y (sln+1), which is a “deformation” of the universal enveloping algebra of the Lie algebra of polynomial maps CI → sln+1. The role of Sl in this theory is played by the degenerate affine Hecke algebra Λl, an algebra whose defining relations are obtained from those of the affine Hecke algebra Ĥl(q 2) by letting q → 1 in a certain non–trivial fashion. 1 Both authors were partially supported by the NSF, DMS-9207701.

Reflections of Root Lattices for Kac–Moody Algebras

In the present paper we determine all the elements in the root lattices of symmetrizable Kac–Moody algebras whose reflections preserve the root systems. Also we discuss elements in the root lattices whose reflections preserve the root lattices.

Generalized Kac-Moody algebras and the modular functionj

The generalized Kac-Moody algebras were introdcued by Borcherds in his study of the vertex algebras and Monstrous Moonshine [B1-B3]. The structure and the representation theory of generalized Kac-Moody algebras are very similar to those of Kac-Moody algebras, and a lot of facts about Kac-Moody algebras can be generalized to generalized Kac-Moody algebras. The main difference is that the generalized Kac-Moody algebras may have imaginm 3, simple roots with norms __ 1. For unitarizable irreducible highest weight modules over symmetrizable generalized Kac-Moody algebras, Borcherds proved a character formula [B2], called the Weyl-Kac-Borcherds formula, which generalizes the WeylKac formula for integrable highest weight modules over symmetrizable Kac-Moody algebras [K]. As in the case of Kac-Moody algebras, the Weyl-Kac-Borcherds formula, when applied to the l-dimensional trivial representation, yields the denominator identio,. In this paper, using the Weyl-Kac-Borcherds formula and the denominator identity, we obtain a closed form root multiplicity formula for all symmetrizable generalized Kac-Moody algebras (Theorem 2.1). Our formula enables us to study the structure of a symmetrizable generalized Kac-Moody algebra 9 as a representation of a Kac-Moody algebra 0o contained in ~. The choice of ~0 gives rise to various expressions of the root multiplicities of 13, which would yield some interesting combinatorial identities. In Sect. 3, we apply this principle to a rank 2 generalized Kac-Moody algebra to obtain a combinatorial identity for the Pascal numbers. In [B3], Borcherds proved the Moonshine conjecture [CNI by constructing the Monster Lie algebra using the Monster vertex algebra [FLM] and the vertex algebra associated with the 2-dimensional even Lorentzian lattice IIl,~ [B 1]. The Monster Lie

Twisted vertex representations of quantum affine algebras

Recent interests in quantum groups are stimulated by their marvelous relations with quantum Yang-Baxter equations, conformal field theory, invariants of links and knots, and q-hypergeometric series. Besides understanding the reason of the appearance of quantum groups in both mathematics and theoretical physics there is a natural problem of finding q-deformations or quantum analogues of known structures. Quantum groups were first defined by Drinfeld [2] and Jimbo [9] (also see [4]) as a q-deformation of the universal enveloping algebras of the KacMoody algebras in the work of trigonometric solutions of Yang-Baxter equations. In the same spirit it was shown in [13], [14], that there exists a 1 1 correspondence between the integrable highest weight representations of symmetrizable Kac-Moody algebras and those of the corresponding quantum groups, where both spaces have the same dimension in the case of generic q (i.e. q is not a root of unity). Moreover, one can be very explicit in the case of quantum gl(n) to write down the irreducible highest weight representations. Quantum affine algebras are the quantum groups associated to affine Lie algebras. Following Drinfeld's realization [-3] of q-analog of loop algebras, the vertex representation of untwisted simply laced quantum affine algebras was constructed in Frenkel-Jing [6], which is a q-deformation of Frenkel-Kac [7] and Segal [15] construction in the theory of affine Lie algebras. Subsequently, the same was done for the quantum affine algebra of type B in [1]. In the present work we construct vertex representations of quantum affine algebras twisted by an automorphism of the Dynkin diagram, which generalizes certain important cases in the ordinary twisted vertex operator calculus [-5,