Yucai Su

Generalized Virasoro and super-Virasoro algebras and modules of the intermediate series

Generalized Virasoro algebras (defined as the universal central extension of some generalized Witt algebras) and super-Virasoro algebras and modules of the intermediate series are studied and discussed.

Classification of Harish-Chandra Modules over the Higher Rank Virasoro Algebras

We classify the Harish-Chandra modules over the higher rank Virasoro and super-Virasoro algebras: It is proved that a Harish-Chandra module, i.e., an irreducible weight module with finite weight multiplicities, over a higher rank Virasoro or super-Virasoro algebra is either a module of the intermediate series, or a finitely-dense module. As an application, it is also proved that an indecomposable weight module with finite weight multiplicities over a generalized Witt algebra is either a uniformly bounded module (i.e., a module with weight multiplicities uniformly bounded) with all nonzero weights having the same multiplicity, or a finitely-dense module, as long as the generalized Witt algebra satisfies one minor condition.

SIMPLE MODULES OVER THE HIGH RANK VIRASORO ALGEBRAS

We obtain that a uniformly bounded simple module over a high rank Virasoro algebra is a module of the intermediate series, and that a simple module with finite dimensional weight spaces is either a module of the intermediate series or a so-called GHW module. *Partly supported by a grant from Shanghai Jiaotong University.

Classification of quasifinite modules over the Lie algebras of Weyl type

Abstract For a nondegenerate additive subgroup Γ of the n-dimensional vector space F n over an algebraically closed field F of characteristic zero, there is an associative algebra and a Lie algebra of Weyl type W (Γ,n) spanned by all differential operators uD1m1⋯Dnmn for u∈ F [Γ] (the group algebra), and m1,…,mn⩾0, where D1,…,Dn are degree operators. In this paper, it is proved that an irreducible quasifinite W ( Z ,1) -module is either a highest or lowest weight module or else a module of the intermediate series; furthermore, a classification of uniformly bounded W ( Z ,1) -modules is completely given. It is also proved that an irreducible quasifinite W (Γ,n) -module is a module of the intermediate series and a complete classification of quasifinite W (Γ,n) -modules is also given, if Γ is not isomorphic to Z .

SECOND COHOMOLOGY GROUP OF GENERALIZED WITT TYPE LIE ALGEBRAS AND CERTAIN REPRESENTATIONS

ABSTRACT We determine the second cohomology groups of Lie algebras of generalized Witt type which are some Lie algebras defined by Passman and Jordan, more general than those defined by Dokovic and Zhao, and slightly more general than those defined by Xu. Among all the 2-cocycles, there is a special one we think interesting. Using this 2-cocycle, we define the so-called Virasoro-like algebras. Then we give a class of their representations.

2-COCYCLES ON THE LIE ALGEBRAS OF GENERALIZED DIFFERENTIAL OPERATORS

ABSTRACT In a recent paper by Zhao and the author, the Lie algebras of Weyl type were defined and studied, where is a commutative associative algebra with an identity element over a field of any characteristic, and is the polynomial algebra of a commutative derivation subalgebra of . In the present paper, the 2-cocycles of a class of the above Lie algebras (which are called the Lie algebras of generalized differential operators in the present paper), with being a field of characteristic 0, are determined. Among all the 2-cocycles, there is a special one which seems interesting. Using this 2-cocycle, the central extension of the Lie algebra is defined.

Harish–Chandra modules of the intermediate series over the high rank Virasoro algebras and high rank super‐Virasoro algebras

Indecomposable modules with weight multiplicities ≤1 over the high rank Virasoro algebras are constructed and classified. A notion of the high rank super‐Virasoro algebras is introduced. The classification of all indecomposable modules with weight multiplicities ≤1 over the high rank super‐Virasoro algebras is obtained.

Classification of harish-chandra modules over the super-virasoro algebras

In this paper, we first construct all indecomposable modules whose dimensions of weight spaces of the even and odd parts are ≤ 1, then classify all Harish-Chandra module over the super-Virasoro algebras, proving that every Harish-Chandra module over the super-Virasoro algebras is either a highest or lowest weight module, or else a module of the intermediate series. This result generalizes a theorem which was originally given as a conjecture by Kac on the Virasoro algebra.

Lie Bialgebras of Generalized WITT Type, II

In an article by Michaelis, a class of infinite-dimensional Lie bialgebras containing the Virasoro algebra was presented. This type of Lie bialgebras was classified by Ng and Taft. In a recent article by Song and Su, Lie bialgebra structures on graded Lie algebras of generalized Witt type with finite dimensional homogeneous components were considered. In this article we consider Lie bialgebra structures on the graded Lie algebras of generalized Witt type with infinite dimensional homogeneous components. By proving that the first cohomology group H1(𝒲, 𝒲 ⊗ 𝒲) is trivial for any graded Lie algebras 𝒲 of generalized Witt type with infinite dimensional homogeneous components, we obtain that all such Lie bialgebras are triangular coboundary.

STRUCTURE OF THE LIE ALGEBRAS RELATED TO THOSE OF BLOCK

Abstract We determine the isomorphism classes of the first family of infinite dimensional simple Lie algebras recently introduced by Xu. The structure space of these algebras is given explicitly. The derivations of these algebras are also determined.