Infinite-dimensional Simple Lie Algebras Research Articles

Jet modules for the centerless Virasoro-like algebra

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a [Formula: see text]-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over [Formula: see text]. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of [Irreducible representations for toroidal Lie algebras, J. Algebras 221 (1999) 188–231; Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004) 23–42]. Consequently, indecomposable jet modules are described using modules over the algebra [Formula: see text], which is the “positive part” of a Block type algebra studied first by [Some infinite-dimensional simple Lie algebras in characteristic [Formula: see text] related to those of Block, J. Pure Appl. Algebra 127(2) (1998) 153–165] and recently by [A [Formula: see text]-graded generalization of the Witt-algebra, preprint; Classification of simple Lie algebras on a lattice, Proc. London Math. Soc. 106(3) (2013) 508–564]).

Representations of four-derivation Lie algebras of Block type

Block introduced certain analogues of the Zassenhaus algebras over a field of characteristic 0. The nongraded infinite-dimensional simple Lie algebras of Block type constructed by Xu can be viewed as generalizations of the Block algebras. In this paper, we construct a family of irreducible modules in terms of multiplication and differentiation operators on “polynomials” for four-devivation nongraded Lie algebras of Block type based on the finite-dimensional irreducible weight modules with multiplicity one of general linear Lie algebras. We also find a new series of submodules from which some irreducible quotient modules are obtained.

Structure of a new class of non-graded infinite-dimensional simple Lie algebras

A new class of infinite-dimensional simple Lie algebras over a field with characteristic 0 are constructed. These are examples of non-graded Lie algebras. The isomorphism classes of these Lie algebras are determined. The structure space of these algebras is given explicitly.

Simple Lie Algebras

Finite and infinite dimensional simple Lie algebras S(A, L) are constructed from a commutative algebra A and nil locally nilpotent Lie subalgebra L of the derivation algebra of A under conditions which are met when A is a domain of characteristic not 2.

STRUCTURE OF THE LIE ALGEBRAS RELATED TO THOSE OF BLOCK

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.

A Class of Infinite Dimensional Simple Lie Algebras

A Class of Infinite Dimensional Simple Lie Algebras

Generalizations of the Block algebras

In this paper, we introduce two new families of infinite-dimensional simple Lie algebras and a new family of infinite-dimensional simple Lie superalgebras. These algebras can be viewed as generalizations of the Block algebras.

Some infinite-dimensional simple Lie algebras in characteristic 0 related to those of Block

Given a nontrivial torsion-free abelian group ( A, +, 0), a field F of characteristic 0, and a nondegenerate bi-additive skew-symmetric map ϑ: A × A → F, we study the Lie algebra L(A, ϑ) over F with basis { e x : x ϵ A⧹{0}} and multiplication [ e x , e y ] = ϑ( x, y) e x + y . We show that L(A, ϑ) is simple, determine its derivations, and show that the locally finite derivations D have the form D( e x ) = μ( x) e x , μ ϵ Hom( A, F). We describe all isomorphisms between two such algebras. Finally, we compute H 2 ( L, F) .

Automorphisms of the Lie Algebras W* in Characteristic 0

In a recent paper [2] we defined four classes of infinite dimensional simple Lie algebras over a field of characteristic 0 which we called W*, S*, H*, and K*. As the names suggest, these classes generalize the Lie algebras of Cartan type. A second paper [3] investigates the derivations of the algebras W* and S*, and the possible isomorphisms between these algebras and the algebras defined by Block [1]. In the present paper we investigate the automorphisms of the algebras of type W*.

On the low-dimensional cohomology of some infinite-dimensional simple Lie algebras

On the low-dimensional cohomology of some infinite-dimensional simple Lie algebras

Finitely presented infinite-dimensional simple Lie algebras

Finitely presented infinite-dimensional simple Lie algebras