Tensor factorization and Spin construction for Kac-Moody algebras

In this paper we discuss the “Factorization phenomenon” which occurs when a representation of a Lie algebra is restricted to a subalgebra, and the result factors into a tensor product of smaller representations of the subalgebra. We analyze this phenomenon for symmetrizable Kac-Moody algebras (including finite-dimensional, semi-simple Lie algebras). We present a few factorization results for a general embedding of a symmetrizable Kac-Moody algebra into another and provide an algebraic explanation for such a phenomenon using Spin construction. We also give some application of these results for semi-simple, finite-dimensional Lie algebras.We extend the notion of Spin functor from finite-dimensional to symmetrizable Kac-Moody algebras, which requires a very delicate treatment. We introduce a certain category of orthogonal g-representations for which, surprisingly, the Spin functor gives a g-representation in Bernstein-Gelfand-Gelfand category O. Also, for an integrable representation, Spin produces an integrable representation. We give the formula for the character of Spin representation for the above category and work out the factorization results for an embedding of a finite-dimensional, semi-simple Lie algebra into its untwisted affine Lie algebra. Finally, we discuss the classification of those representations for which Spin is irreducible.     

Differential equations in vertex algebras and simple modules for the Lie algebra of vector fields on a torus

We study irreducible representations for the Lie algebra of vector fields on a 2-dimensional torus constructed using the generalized Verma modules. We show that for a certain choice of parameters these representations remain irreducible when restricted to a loop subalgebra in the Lie algebra of vector fields. We prove this result by studying vertex algebras associated with the Lie algebra of vector fields on a torus and solving non-commutative differential equations that we derive using the vertex algebra technique.     

The socle of a nondegenerate Lie algebra

We define the socle of a nondegenerate Lie algebra as the sum of all its minimal inner ideals. The socle turns out to be an ideal which is a direct sum of simple ideals, and satisfies the descending chain condition on principal inner ideals. Every classical finite dimensional Lie algebra coincides with its socle, while relevant examples of infinite dimensional Lie algebras with nonzero socle are the simple finitary Lie algebras and the classical Banach Lie algebras of compact operators on an infinite dimensional Hilbert space. This notion of socle for Lie algebras is compatible with the previous ones for associative algebras and Jordan systems. We conclude with a structure theorem for simple nondegenerate Lie algebras containing abelian minimal inner ideals, and as a consequence we obtain that a simple Lie algebra over an algebraically closed field of characteristic 0 is finitary if and only if it is nondegenerate and contains a rank-one element.     

Irreducible Lie-Yamaguti algebras

Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their Lie inner derivation algebra are the algebraic counterpart of the isotropy irreducible homogeneous spaces.These systems will be shown to split into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types will be classified and most of them will be shown to be related to a Generalized Tits Construction of Lie algebras.     

Triangulated categories and Kac-Moody algebras

By using the Ringel-Hall algebra approach, we find a Lie algebra arising in each triangulated category with T ²=1, where T is the translation functor. In particular, the generic form of the Lie algebras determined by the root categories, the 2-period orbit categories of the derived categories of finite dimensional hereditary associative algebras, gives a realization of all symmetrizable Kac-Moody Lie algebras. Oblatum 4-XII-1998 & 11-XI-1999?Published online: 21 February 2000     

Finite vertex algebras and nilpotence

I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (= without nilpotent elements) finite vertex algebra is nilpotent.     

Tubular algebras and affine Kac-Moody algebras

The purpose of this paper is to construct quotient algebras L(A)1C/I(A) of complex degenerate composition Lie algebras L(A)1C by some ideals, where L(A)1C is defined via Hall algebras of tubular algebras A, and to prove that the quotient algebras L(A)1C/I(A) are isomorphic to the corresponding affine Kac-Moody algebras. Moreover, it is shown that the Lie algebra Lre(A)1C generated by A-modules with a real root coincides with the degenerate composition Lie algebra L(A)1C generated by simple A-modules.     

Affine Jacquet functors and Harish-Chandra categories

We define an affine Jacquet functor and use it to describe the structure of induced affine Harish-Chandra modules at noncritical levels, extending the theorem of Kac and Kazhdan on the structure of Verma modules in the Bernstein-Gelfand-Gelfand categories O for Kac-Moody algebras. This is combined with a vanishing result for certain extension groups to construct a block decomposition of the categories of affine Harish-Chandra modules of Lian and Zuckerman. The latter provides an extension of the works of Rocha-Caridi and Wallach [A. Rocha-Caridi, N.R. Wallach, Projective modules over infinite dimensional graded Lie algebras, Math. Z. 180 (1982) 151-177] and Deodhar, Gabber and Kac [V. Deodhar, O. Gabber, V. Kac, Structure of some categories of representations of infinite-dimensional Lie algebras, Adv. Math. 45 (1982) 92-116] on block decompositions of BGG categories for Kac-Moody algebras. We also derive a compatibility relation between the affine Jacquet functor and the Kazhdan-Lusztig tensor product and apply it to prove that the affine Harish-Chandra category is stable under fusion tensoring with the Kazhdan-Lusztig category. This compatibility will be further applied in studying translation functors for the affine Harish-Chandra category, based on the fusion tensor product.     

On the simplicity of some kac-moody groups

Recently, Marcuson extended the classical construction of Tits systems in Steinberg groups to include the Kac-Moody Steinberg groups associated with the infinite dimensional versions of the great Lie algebras. If these Lie algebras and their Kac-Moody groups are viewed as limits of their finite dimensional counterparts, more direct methods may be employed. In fact, the Kac-Moody Chevalley groups of these Lie algebras are seen to be simple.     

Modified regular representations of affine and Virasoro algebras, VOA structure and semi-infinite cohomology

We find a counterpart of the classical fact that the regular representation R(G) of a simple complex group G is spanned by the matrix elements of all irreducible representations of G. Namely, the algebra of functions on the big cell G₀⊂G of the Bruhat decomposition is spanned by matrix elements of big projective modules from the category O of representations of the Lie algebra g of G, and has the structure of a g⊕g-module.The standard regular representation of the affine group has two commuting actions of the Lie algebra with total central charge 0, and carries the structure of a conformal field theory. The modified versions and , originating from the loop version of the Bruhat decomposition, have two commuting -actions with central charges shifted by the dual Coxeter number, and acquire vertex operator algebra structures derived from their Fock space realizations, given explicitly for the case G=SL(2,C).The quantum Drinfeld-Sokolov reduction transforms the representations of the affine Lie algebras into their W-algebra counterparts, and can be used to produce analogues of the modified regular representations. When g=sl(2,C) the corresponding W-algebra is the Virasoro algebra. We describe the Virasoro analogues of the modified regular representations, which are vertex operator algebras with the total central charge equal to 26.The special values of the total central charges in the affine and Virasoro cases lead to the non-trivial semi-infinite cohomology with coefficients in the modified regular representations. The inherited vertex algebra structure on this cohomology degenerates into a supercommutative associative superalgebra. We describe these superalgebras in the case when the central charge is generic, and identify the 0th cohomology with the Grothendieck ring of finite-dimensional G-modules. We conjecture that for the integral values of the central charge the 0th semi-infinite cohomology coincides with the Verlinde algebra and its counterpart associated with the big projective modules.     

D_4型广义扭仿射李代数及其模

本文是[3]的继续,将讨论D4型的广义扭仿射李代数及其表示理论;证明作用在其不可约模上的一类算子的局部幂零性.     

The centroid of extended affine and root graded Lie algebras

We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.     

Bases, filtrations and module decompositions of free Lie algebras

We use the technique known as elimination to devise some new bases of the free Lie algebra which (like classical Hall bases) consist of Lie products of left normed basic Lie monomials. Our bases yield direct decompositions of the homogeneous components of the free Lie algebra with direct summands that are particularly easy to describe: they are tensor products of metabelian Lie powers. They also give rise to new filtrations and decompositions of free Lie algebras as modules for groups of graded algebra automorphisms. In particular, we obtain some new decompositions for free Lie algebras and free restricted Lie algebras over fields of positive characteristic.     

Elliptic Lie algebras and tubular algebras

This article is to study relations between tubular algebras of Ringel and elliptic Lie algebras in the sense of Saito-Yoshii. Using the explicit structure of the derived categories of tubular algebras given by Happel-Ringel, we prove that the elliptic Lie algebra of type , , or is isomorphic to the Ringel-Hall Lie algebra of the root category of the tubular algebra with the same type. As a by-product of our proof, we obtain a Chevalley basis of the elliptic Lie algebra following indecomposable objects of the root category of the corresponding tubular algebra. This can be viewed as an analogue of the Frenkel-Malkin-Vybornov theorem in which they described a Chevalley basis for each untwisted affine Kac-Moody Lie algebra by using indecomposable representations of the corresponding affine quiver.     

Nonlocal vertex algebras generated by formal vertex operators

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundations for this study. For any vector space W, we study what we call quasi compatible subsets of Hom (W,W((x))) and we prove that any maximal quasi compatible subspace has a natural nonlocal (namely noncommutative) vertex algebra structure with W as a natural faithful quasi module in a certain sense, and that any quasi compatible subset generates a nonlocal vertex algebra with W as a quasi module. In particular, taking W to be a highest weight module for a quantum affine algebra we obtain a nonlocal vertex algebra with W as a quasi module. We also formulate and study a notion of quantum vertex algebra and we give general constructions of nonlocal vertex algebras, quantum vertex algebras and their modules.     

On Borcherds type Lie algebras

For each even lattice \({\mathcal L}\), there is a canonical way to construct an infinite-dimensional Lie algebra via lattice vertex operator algebra theory, we call this Lie algebra and its subalgebras the Borcherds type Lie algebras associated to \({\mathcal L}\). In this paper, we apply this construction to even lattices arising from representation theory of finite-dimensional associative algebras. This is motivated by the different realizations of Kac-Moody algebras by Borcherds using lattice vertex operators and by Peng-Xiao using Ringel-Hall algebras respectively. For any finite-dimensional algebra \(A\) of finite global dimension, we associate a Borcherds type Lie algebra \(\mathfrak {BL}(A)\) to \(A\). In contrast to the Ringel-Hall Lie algebra approach, \(\mathfrak {BL}(A)\) only depends on the symmetric Euler form or Tits form but not the full representation theory of \(A\). However, our results show that for certain classes of finite-dimensional algebras whose representation theory is ’controlled’ by the Euler bilinear forms or Tits forms, their Borcherds type Lie algebras do have close relations with the representation theory of these algebras. Beyond the class of hereditary algebras, these algebras include canonical algebras, representation-directed algebras and incidence algebras of finite prinjective types.     

Motivic zeta functions of infinite-dimensional Lie algebras

The paper shows how to associate a motivic zeta function with a large class of infinite dimensional Lie algebras. These include loop algebras, affine Kac-Moody algebras, the Virasoro algebra and Lie algebras of Cartan type. The concept of a motivic zeta functions provides a good language to talk about the uniformity in p of local p-adic zeta functions of finite dimensional Lie algebras. The theory of motivic integration is employed to prove the rationality of motivic zeta functions associated to certain classes of infinite dimensional Lie algebras.     

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     

Representations of a class of lattice type vertex algebras

In this paper we study the representation theory for certain “half lattice vertex algebras.” In particular we construct a large class of irreducible modules for these vertex algebras. We also discuss how the representation theory of these vertex algebras are related to the representation theory of some associative algebras.     

Irreducible Lie-Yamaguti algebras of generic type

Lie-Yamaguti algebras (or generalized Lie triple systems) are binary-ternary algebras intimately related to reductive homogeneous spaces. The Lie-Yamaguti algebras which are irreducible as modules over their inner derivation algebras are the algebraic counterparts of the isotropy irreducible homogeneous spaces.These systems splits into three disjoint types: adjoint type, non-simple type and generic type. The systems of the first two types were classified in a previous paper through a generalized Tits Construction of Lie algebras. In this paper, the Lie-Yamaguti algebras of generic type are classified by relating them to several other nonassociative algebraic systems: Lie and Jordan algebras and triple systems, Jordan pairs or Freudenthal triple systems.