Derivation-Simple Algebras and the Structures of Lie Algebras of Witt Type

Abstract

We classify all the pairs of a commutative associative algebra with an identity element and its finite-dimensional locally finite Abelian derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra over an algebraically closed field with characteristic 0. Such pairs are the fundamental ingredients for constructing simple Lie algebras of Cartan type. Moreover, we determine the isomorphism classes of the simple Lie algebras of Witt type. The structure space of these algebras is given explicitly.

Structures of Not-finitely Graded Lie Algebras Related to Generalized Virasoro Algebras

“…Thus, W (Γ) appears very naturally. One can observe that this algebra looks very similar to the Lie algebra W = W (0, 1, 0; Γ) of Witt type studied in [8], which has the same basis with bracket [L α,i , L β,j ] = (α − β)L α+β,i+j + (j − i)L α+β,i+j−1 .…”

“…Various generalizations of the Virasoro algebra have been studied by several authors (e.g., [3][4][5][8][9][10][11][12]). In this paper, (1+t) i , a ∈ Γ, i ∈ Z + .…”

Structure of the Lie Algebras Related to Those of Block

“…(5), (8), (9). In this case, the structure space is the set which is the union of two copies of M 3 and two copies of M 4 .…”

“…Such pairs (A, D) with D being locally finite were classified in Ref. (8). Based on this classification it was very natural for Xu (12) to give generalizations of Block algebras.…”

Structure of Algebras of Weyl Type

“…From Su et al (2000), we know that the pairs (A, D), where A is a commutative associative algebra with an identity element 1 over F and D is a nonzero finite dimensional F-vector space of locally finite commuting F-derivations of A such that A is D-simple, are essentially those constructed as follows.…”