On regular 𝐺-gradings

Aljadeff, Eli; David, Ofir

doi:10.1090/S0002-9947-2014-06200-4

On regular $G$-gradings
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by Eli Aljadeff and Ofir David PDF

Trans. Amer. Math. Soc. 367 (2015), 4207-4233 Request permission

Abstract:

Let $A$ be an associative algebra over an algebraically closed field $\mathbb {F}$ of characteristic zero and let $G$ be a finite abelian group. Regev and Seeman introduced the notion of a regular $G$-grading on $A$, namely a grading $A=\bigoplus _{g\in G}A_{g}$ that satisfies the following two conditions: $(1)$ for every integer $n\geq 1$ and every $n$-tuple $(g_{1},g_{2},\dots ,g_{n})\in G^{n}$, there are elements, $a_{i}\in A_{g_{i}}$, $i=1,\dots ,n$, such that $\prod _{1}^{n}a_{i}\neq 0$; $(2)$ for every $g,h\in G$ and for every $a_{g}\in A_{g},b_{h}\in A_{h}$, we have $a_{g}b_{h}=\theta _{g,h}b_{h}a_{g}$ for some nonzero scalar $\theta _{g,h}$. Then later, Bahturin and Regev conjectured that if the grading on $A$ is regular and minimal, then the order of the group $G$ is an invariant of the algebra. In this article we prove the conjecture by showing that $ord(G)$ coincides with an invariant of $A$ which appears in PI theory, namely $exp(A)$ (the exponent of $A$). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.

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