Simple associative algebras with finite Z-grading

Oleg N. Smirnov

Abstract

The aim of this paper is to describe finite Z-gradings of simple associative algebras. Our description has an especially simple form for unital algebras. In this case we show that any such grading R = Å_{i = -n}ⁿ R_i arises from the Peirce decomposition of the algebra with respect to a complete system of orthogonal idempotents E = {e₀,e₁,...,e_n} as follows:

R_i =

å
p-q = i

e_pRe_q for i = -n,...,n .

In the general case we prove that any simple Z-graded algebra is a generalized matrix algebra in sense of G. M. Bergman, i.e., R = Å_{p,q = 0}ⁿ R_p,q with multiplication R_p,qR_s,t Í d_q,sR_p,t, and the grading of R is induced by this decomposition, namely

R_i =

å
p-q = i

R_p,q for i = -n,...,n .

As a corollary of this description we obtain that any simple Lie algebra from a certain class of Lie algebras containing the class of finite dimensional simple Lie algebras over a field of characteristic 0 has a Z-grading with at most 5 summands. This fact in turn allows one to realize these algebras as a generalized Tits-Kantor-Koecher construction.

Original Article in DVI Format