Crossed modules of Lie color algebras

The linear operations of the equivalent classes of crossed modules of Lie color algebras are studied. The set of the equivalent classes of crossed modules is proved to be a vector space, which is isomorphic with the homogeneous components of degree zero of the third cohomology group of Lie color algebras. As an application of this theory, the crossed modules of Witt type Lie color algebras is described, and the result is proved that there is only one equivalent class of the crossed modules of Witt type Lie color algebras when the abelian group Г is equal to Г＋. Finally, for a Witt type Lie color algebra, the classification of its crossed modules is obtained by the isomorphism between the third cohomology group and the crossed modules.

Borel Subalgebras of the Witt Algebra

Let F be an algebraically closed field of characteristic p 〉 3, and g be the Witt algebra over F. Let N be the nilpotent cone of g. An explicit description of N is given, so that the conjugacy classes of Borel subalgebras of g under the automorphism group of g are determined. In contrast with only one conjugacy class of Borel subalgebras in a classical simple Lie algebra, there are two conjugacy classes of Borel subalgebras in g. The representatives of conjugacy classes of Borel subalgebras, i.e.,the so-called standard Borel subalgebras, are precisely given.

Infinite-dimensional 3-Lie algebras and their ：onnections to Harish-Chandra modules

We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.

Commuting variety of Witt algebra

Let g = W1 be the Witt algebra over an algebraically closed field k of characteristic p 〉 3, and let ∮（g） = {（x,y） ∈ g×g [x,y] = 0} be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250： 473-484], we show that the variety ∮（g） is reducible, and not equidimensional. Irreducible components of ∮（g） and their dimensions are precisely given. As a consequence, the variety ∮（g） is not normal.

Derivations and Automorphism Group of Original Deformative Schrodinger-Virasoro Algebra

In this paper, we determine the derivation algebra and the automorphism group of the original deformative Schrodinger-Virasoro algebra, which is the semi-direct product Lie algebra of the Witt algebra and its tensor density module Ig（a, b）.

ENDOMORPHISMS OF LIE ALGEBRA F[t]a/dt

Let F be a field of characteristic zero. W_n = F[t_1^(+-1), t_2^(+-1), ...,t_n^(+-1)] (partial deriv)/((partial deriv)t_1) + ... + F[t_1^(+-1), t_2^(+-1), ..., t_n^(+-1)](partial deriv)/((partial deriv)t_n) is the Witt algebra over F, W_n^+ = F[t_1, t_2 ..., t_n](partial deriv)/((partial deriv)t_1) + ... + F[t_1, t_2 ..., t_n] (partial deriv)/((partialderiv)t_n) is Lie subalgebra of W_n. It is well known both W_n and W_n^+ are simple infinitedimensional Lie algebra. In Zhao's paper, it was conjectured that End(W_n^+) - {0} = Aut(W_n^+) andit was proved that the validity of this conjecture implies the validity of the well-known Jacobianconjecture. In this short note, we check the conjecture above for n = 1. We show End(W_1^+) - {0} =Aut(W_1^+).

The σ-Derivations of C[x^±1, y±1]

In the present paper, the σ-derivations of C[x^±1, y±1] are determined, and their applications are considered.