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.     

Affine and degenerate affine BMW algebras: actions on tensor space

The affine and degenerate affine Birman–Murakami–Wenzl (BMW) algebras arise naturally in the context of Schur–Weyl duality for orthogonal and symplectic quantum groups and Lie algebras, respectively. Cyclotomic BMW algebras, affine and cyclotomic Hecke algebras, and their degenerate versions are quotients of the affine and degenerate affine BMW algebras. In this paper, we explain how the affine and degenerate affine BMW algebras are tantalizers (tensor power centralizer algebras) by defining actions of the affine braid group and the degenerate affine braid algebra on tensor space and showing that, in important cases, these actions induce actions of the affine and degenerate affine BMW algebras. We then exploit the connection to quantum groups and Lie algebras to determine universal parameters for the affine and degenerate affine BMW algebras. Finally, we show that the universal parameters are central elements—the higher Casimir elements for orthogonal and symplectic enveloping algebras and quantum groups.     

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

The Core of a Locally Extended Affine Lie Algebra

Locally extended affine Lie algebras are a general version of extended affine Lie algebras. In this article, we completely describe the structure of the core of a locally extended affine Lie algebra. We prove that the core of a locally extended affine Lie algebra is a direct limit of Lie tori.     

Extended affine Lie superalgebras and their affinizations

Extended affine Lie superalgebras are super versions of the defining axioms of extended affine Lie algebras or more generally invariant affine reflection algebras. This class includes finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras. In this paper, an affinization process is introduced for the class of extended affine Lie superalgebras, and the necessary conditions for an extended affine Lie superalgebra to be invariant under this process are presented. Moreover, new extended affine Lie superalgebras are constructed by means of the affinization process.     

Direct Unions of Lie Tori (Realization of Locally Extended Affine Lie Algebras)

In this work, we consider realizations of locally extended affine Lie algebras, in the level of core modulo center. We provide a framework similar to the one for extended affine Lie algebras by “direct unions.” Our approach suggests that the direct union of existing realizations of extended affine Lie algebras, in a rigorous mathematical sense, would lead to a complete realization of locally extended affine Lie algebras, in the level of core modulo center. As an application of our results, we realize centerless cores of locally extended affine Lie algebras with specific root systems of types A₁, B, C, and BC.     

Affine Vogan diagrams and symmetric pairs

We introduce the affine Vogan diagrams of complex simple Lie algebras. These are generalizations of Vogan diagrams, and we study the involutions represented by them. We apply these diagrams to study the symmetric pairs, in particular the associated and symplectic symmetric pairs.     

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.     

A Generalization of Extended Affine Lie Algebras

We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and extended affine Lie algebras. Our results generalize well-known properties of these examples.     

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.     

Lie algebras with quadratic dimension equal to 2

The quadratic dimension of a Lie algebra is defined as the dimension of the linear space spanned by all its invariant non-degenerate symmetric bilinear forms. We prove that a quadratic Lie algebra with quadratic dimension equal to 2 is a local Lie algebra, this is to say, it admits a unique maximal ideal. We describe local quadratic Lie algebras using the notion of double extension and characterize those with quadratic dimension equal to 2 by the study of the centroid of such Lie algebras. We also give some necessary or sufficient conditions for a Lie algebra to have quadratic dimension equal to 2. Examples of local Lie algebras with quadratic dimension larger than 2 are given.     

On abelian subalgebras and ideals of maximal dimension in supersolvable Lie algebras

In this paper, the main objective is to compare the abelian subalgebras and ideals of maximal dimension for finite-dimensional supersolvable Lie algebras. We characterise the maximal abelian subalgebras of solvable Lie algebras and study solvable Lie algebras containing an abelian subalgebra of codimension 2. Finally, we prove that nilpotent Lie algebras with an abelian subalgebra of codimension 3 contain an abelian ideal with the same dimension, provided that the characteristic of the underlying field is not 2. Throughout the paper, we also give several examples to clarify some results.     

Blocks and modules for Whittaker pairs

Inspired by recent activities on Whittaker modules over various (Lie) algebras, we describe a general framework for the study of Lie algebra modules locally finite over a subalgebra. As a special case, we obtain a very general set-up for the study of Whittaker modules, which includes, in particular, Lie algebras with triangular decomposition and simple Lie algebras of Cartan type. We describe some basic properties of Whittaker modules, including a block decomposition of the category of Whittaker modules and certain properties of simple Whittaker modules under some rather mild assumptions. We establish a connection between our general set-up and the general set-up of Harish-Chandra subalgebras in the sense of Drozd, Futorny and Ovsienko. For Lie algebras with triangular decomposition, we construct a family of simple Whittaker modules (roughly depending on the choice of a pair of weights in the dual of the Cartan subalgebra), describe their annihilators, and formulate several classification conjectures. In particular, we construct some new simple Whittaker modules for the Virasoro algebra. Finally, we construct a series of simple Whittaker modules for the Lie algebra of derivations of the polynomial algebra, and consider several finite-dimensional examples, where we study the category of Whittaker modules over solvable Lie algebras and their relation to Koszul algebras.     

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.     

广义仿射李代数上的非交换Poisson代数结构

非交换的Poisson代数同时具有(未必交换的)结合代数和李代数两种代数结构,且结合代数和李代数之间满足所谓的Leibniz法则.本文确定了一般广义仿射李代数上所有的Poisson代数结构.     

Infinitesimal deformations of restricted simple Lie algebras II

We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartan-type: the Contact and the Hamiltonian Lie algebras.     

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.