Simple Lie Color Algebras of Weyl Type

Details Simple Lie Color Algebras of Weyl Type Simple Lie Color Algebras of Weyl Type

2003-04-05

arxiv.org/abs/math/0304075v1

Mathematics

Quantum Algebra

10 pages, latex; to appear in Israel J. Math

Scientific paper

For an $(\epsilon,G)$-color-commutative associative algebra $A$ with an identity element over a field $F$ of characteristic not 2, and for a color-commutative subalgebra $D$ of color-derivations of $A$, denote by $A[D]$ the associative subalgebra of ${\rm End}(A)$ generated by $A$ (regarding as operators on $A$ via left multiplication) and $D$. It is easily proved that, as an associative algebra, $A[D]$ is $G$-graded simple if and only if $A$ is $\G$-graded $D$-simple. Suppose $A$ is $\G$-graded $D$-simple. Then, (a) $A[D]$ is a free left $A$-module; (b) as a Lie color algebra, the subquotient $[A[D],A[D]]/Z(A[D])\cap[A[D],A[D]]$ is simple (except one minor case), where $Z(A[D])$ is the color center of $A[D]$. The structure of this subquotient is explicitly described.

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