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On regular $ G$-gradings


Authors: Eli Aljadeff and Ofir David
Journal: Trans. Amer. Math. Soc. 367 (2015), 4207-4233
MSC (2010): Primary 16R99, 16W50
DOI: https://doi.org/10.1090/S0002-9947-2014-06200-4
Published electronically: December 5, 2014
MathSciNet review: 3324925
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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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Additional Information

Eli Aljadeff
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Email: aljadeff@tx.technion.ac.il

Ofir David
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Email: ofirdav@tx.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9947-2014-06200-4
Received by editor(s): March 6, 2013
Received by editor(s) in revised form: March 7, 2013, and May 29, 2013
Published electronically: December 5, 2014
Additional Notes: The first author was partially supported by the Israel Science Foundation (grant No. 1283/08 and grant No. 1017/12) and by the Glasberg-Klein Research Fund.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.