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.     

非扭仿射Kac-Moody代数中Borel子代数的扩代数

决定了非扭仿射Kac-Moody代数中所有包含标准Borel子代数的子代数。     

Majid conjecture: quantum Kac-Moody algebras version

Based on the n-fold tensor product version of the generalized double-bosonization construction, we prove the Majid conjecture of the quantum Kac-Moody algebras version. Particularly, we give explicitly the double-bosonization type-crossing constructions of quantum Kac-Moody algebras for affine types G_2^{(1)} , E_2^{(1)},and Tp,q,r, and in this way, we can recover generators of quantum Kac-Moody algebras with braided groups defined by R-matrices in the related braided tensor category. This gives us a better understanding for the algebra structures themselves of the quantum Kac-Moody algebras as a certain extension of module-algebras/module-coalgebras with respect to the related quantum subalgebras of finite types inside.     

Varieties of affine Kac-Moody algebras

In this paper the identities of the complex affine Kac-Moody algebras are studied. It is proved that the identities of twisted affine algebras coincide with those of the corresponding nontwisted algebras. Moreover, in the class of nontwisted affine Kac-Moody algebras, each of these algebras is uniquely defined by its identities. It is shown that the varieties of affine algebras, as well as the varieties defined by finitely generated three-step solvable Lie algebras, have exponential growth. Translated fromMatematicheskie Zametki, Vol. 62 No. 1, pp. 95–102, July 1997. Translated by A. I. Shtern     

Verma modules over generalized Witt algebras

We constuct and investigate a structure of Verma-like modules over generalized Witt algebras. We also prove Futorny-like theorem for irreducible weight modlues whose dimensions of the weight spaces are uniformly bounded.     

Fock Space Realizations of Imaginary Verma Modules

This work expands to the setting of the results of H. Jakobsen and V. Kac and independently D. Bernard and G. Felder on the realization of , in terms of infinite sums of partial differential operators. We note in the paper that, in the generic case, these geometric constructions are just realizations of the imaginary Verma modules studied by V. Futorny. Presented by A. VerschorenMathematics Subject Classifications (2000) Primary: 17B67, 81R10.     

Submodule Structure of Generalized Verma Modules Induced from Generic Gelfand-Zetlin Modules

For complex Lie algebra sl(n, C) we study the submodule structure of generalized Verma modules induced from generic Gelfand-Zetlin modules over some subalgebra of type sl(k, C). We obtain necessary and sufficient conditions for the existence of a submodule generalizing the Bernstein-Gelfand-Gelfand theorem for Verma modules.     

Generalized Verma modules over some Block algebras

In this paper, a class of generalized Verma modules M(V) over some Block type Lie algebra ℬ(G) are constructed, which are induced from nontrivial simple modules V over a subalgebra of ℬ(G). The irreducibility of M(V) is determined.      

On derived categories of modules over 2-vertex basic algebras

We investigate finite dimensional 2-vertex basic algebras of finite global dimension and the derived categories of modules over such algebras. We prove that any superrigid object in the derived category of modules over a “loop-kind” two-vertex algebra is a pure module up to the action of Serre functor and translation. All superrigid objects in the derived categories of modules over two-vertex algebras of global dimension 2 are described. Also we obtain a complete classification of two-vertex basic algebras possessing a full exceptional pair in the derived category of modules.     

Verma modules for twisted Yangians

Abstract

A sufficient and necessary condition for the reducibility of the Verma modules over the twisted Yangians of rank one is given.     

Quasi-Whittaker modules over the conformal Galilei algebras

In this paper, we study quasi-Whittaker modules over the conformal Galilei algebras. We determine all quasi-Whittaker vectors and the simplicities of the universal quasi-Whittaker modules. For the reducible ones, we determine the corresponding simple quotient modules. Thus we classify all simple quasi-Whittaker modules for the conformal Galilei algebras.     

Representations of Loop Kac-Moody Lie Algebras

We study representations of the Loop Kac-Moody Lie algebra 𝔤 ?A, where 𝔤 is any Kac-Moody algebra and A is a ring of Laurent polynomials in n commuting variables. In particular, we study representations with finite dimensional weight spaces and their graded versions. When we specialize 𝔤 to be a finite dimensional or affine Lie algebra we obtain modules for toroidal Lie algebras.     

Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field F of characteristic zero, a Lie algebra B（G） of Block type is defined with basis {Lα,i｜ α∈G, i∈Z＋} and relations [Lα,i, Lβ,j] = （β-α）Lα＋β,i＋j＋（αj-βi）Lα＋β,Lα＋β,i＋j-1.It is proved that an irreducible highest weight B（Z）-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order λ on G and any ∧∈B（G）0^＊（the dual space of B（G）0 = span{L0,i｜i∈Z＋}）, a Verma B（G）-module M（∧,λ） is defined, and the irreducibility of M（A,λ） is completely determined.     

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.

     

Kac-Moody群上可微分模的刻画

对Ｋａｙ－Ｍｏｏｄｙ代数ｇ′上的任意可积模（Ｖ，ｄπ），通过指数可以把它提升为同ｇ′关联的Ｋａｃ－Ｍｏｏｄｙ群Ｇ上的模（Ｖ，π），Ｇ上的这种模称为可微分模．本文将刻画Ｇ上的可微分模并且证明，模（Ｖ，π）是可微分模当且仅当Ｖ到每个根子群Ｕ_ａ的限制都是Ｕ_ａ的一个有理表示．依据这种刻画，得到一个有趣的结果：有理数域Ｑ上的Ｃｈｅｖａｌｅｙ群Ｇ（Ｑ）的所有有限维模都是可微分模     

Existence of critical modules of GK-dimension 2 over elliptic algebras

We show that over an elliptic algebra, critical modules of Gelfand-Kirillov dimension 2 exist in all multiplicities (assuming the ground field is uncountable, algebraically closed). Geometrically, this shows that in a quantum plane there exist ``irreducible curve" modules of all possible degrees.

     

量子广义Kac-Moody代数的整形式

主要讨论一阶量子广义Kac-Moody代数U_q(2α)的结构,其中a∈Z0.在此基础上,刻画了量子广义代数U_q(g)的另一种整形式.