Verma type modules of level zero for affine Lie algebras

We study the structure of Verma type modules of level zero induced from non-standard Borel subalgebras of an affine Kac-Moody algebra. For such modules in ``general position' we describe the unique irreducible quotients, construct a BGG type resolution and prove the BGG duality in certain categories. All results are extended to generalized Verma type modules of zero level.

     

Vertex operator algebras and representations of affine Lie algebras

We announce the construction of an explicit basis for all integrable highest weight modules over the Lie algebra A ₁ ⁽¹⁾. The construction uses representations of vertex operator algebras and leads to combinatorial identities of Rogers-Ramanujan-type.     

Graded Lie algebras, representation theory, integrable mappings, and integrable systems

A new class of integrable mappings and chains is introduced. The corresponding 1+2 integrable systems that are invariant under such integrable mappings are presented in an explicit form. Soliton-type solutions of these systems are constructed in terms of matrix elements of fundamental representations of semisimple A_n algebras for a given group element. The possibility of generalizing this construction to the multidimensional case is discussed. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 251–271, February, 2000.     

Vertex operator algebras associated to modified regular representations of affine Lie algebras

Let G be a simply-connected complex Lie group with simple Lie algebra g and let be its affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of N-graded vertex operator algebras (VOAs) associated to g. These vertex operator algebras contain the algebra of regular functions on G as the conformal weight 0 subspaces and are -modules of dual levels in the sense that , where h^∨ is the dual Coxeter number of g. This family of VOAs was previously studied by Arkhipov-Gaitsgory and Gorbounov-Malikov-Schechtman from different points of view. We show that when k is irrational, the vertex envelope of the vertex algebroid associated to G and the level k is isomorphic to the vertex operator algebra we constructed above. The case of rational levels is also discussed.     

Elliptic affine Lie algebras

G. M. Krzhizhanovskii Energetics Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 24, No. 3, pp. 51–61, July–September, 1990.     

Graded subalgebras of affine Kac-Moody algebras

We determine the maximal graded subalgebras of affine Kac-Moody algebras. We also show that the maximal graded subalgebras of loop algebras are essentially loop algebras. Supported by the Binational Science Foundation United States — Israel, Grant No. 92-00034.     

Modules for double affine Lie algebras

Imaginary Verma modules, parabolic imaginary Verma modules,and Verma modules at level zero for double affine Lie algebras are constructed using three different triangular decompositions. Their relations are investigated,and several results are generalized from the affine Lie algebras. In particular,imaginary highest weight modules, integrable modules, and irreducibility criterion are also studied.     

Quantum affine algebras and their representations

Some questions on the representation theory of quantum affine algebras are discussed from the categorical point of view.     

Graded Lie algebras with finite polydepth

If A is a graded connected algebra then we define a new invariant, polydepthA, which is finite if for some A-module M of at most polynomial growth. Theorem 1: If f:X→Y is a continuous map of finite category, and if the orbits of acting in the homology of the homotopy fibre grow at most polynomially, then has finite polydepth. Theorem 5: If L is a graded Lie algebra and polydepthUL is finite then either L is solvable and UL grows at most polynomially or else for some integer d and all r, ∑_i=k+1^k+ddimL_i?k^r, k? some k(r).     

Varieties of representations of Lie algebras

Riga. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 2, pp. 144–153, March–April, 1988.     

The degeneracy of extended affine Lie algebras

We construct degenerate extended affine Lie algebras from a given nondegenerate extended affine Lie algebra and show that all degenerate extended affine Lie algebras are obtained in this way. Received: 21 January 1997