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arXiv:1002.1385v1[math.RA]6Feb2010 GROUP GRADED PI-ALGEBRAS AND THEIR CODIMENSION
 

Summary: arXiv:1002.1385v1[math.RA]6Feb2010
GROUP GRADED PI-ALGEBRAS AND THEIR CODIMENSION
GROWTH
ELI ALJADEFF
Abstract. Let W be an associative PI- algebra over a field F of characteristic
zero. Suppose W is G-graded where G is a finite group. Let exp(W ) and
exp(We) denote the codimension growth of W and of the identity component
We, respectively. The following inequality had been conjectured by Bahturin
and Zaicev: exp(W ) |G|2 exp(We). The inequality is known in case the
algebra W is affine (i.e. finitely generated). Here we prove the conjecture in
general.
1. Introduction
Let W be any PI-algebra over a field of characteristic zero. Suppose W is G-
graded where G is any finite group. Let exp(W) and exp(We) be the exponents of
the algebra W and of its e-component (determined by the G-grading). The main
objective of this paper is to prove the following conjecture (see [8]):
Conjecture 1.1.
exp(W) | G |2
exp(We).
The case where W is affine was proved in [1]. Here we show that the conjecture

  

Source: Aljadeff, Eli - Department of Mathematics, Technion, Israel Institute of Technology

 

Collections: Mathematics