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Graded identities of matrix algebras and the universal graded algebra


Authors: Eli Aljadeff, Darrell Haile and Michael Natapov
Journal: Trans. Amer. Math. Soc. 362 (2010), 3125-3147
MSC (2000): Primary 16W50, 16R10, 16R50; Secondary 16S35, 16K20
DOI: https://doi.org/10.1090/S0002-9947-10-04811-7
Published electronically: January 7, 2010
MathSciNet review: 2592949
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Abstract: We consider fine group gradings on the algebra $ M_n(\mathbb{C})$ of $ n$ by $ n$ matrices over the complex numbers and the corresponding graded polynomial identities. Given a group $ G$ and a fine $ G$-grading on $ M_n(\mathbb{C})$, we show that the $ T$-ideal of graded identities is generated by a special type of identity, and, as a consequence, we solve the corresponding Specht problem for this case. Next we construct a universal algebra $ U$ (depending on the group $ G$ and the grading) in two different ways: one by means of polynomial identities and the other one by means of a generic two-cocycle (this parallels the classical constructions in the nongraded case). We show that a suitable central localization of $ U$ is Azumaya over its center and moreover, its homomorphic images are precisely the $ G$-graded forms of $ M_n(\mathbb{C})$. Finally, we consider the ring of central quotients of $ U$ which is a central simple algebra over the field of quotients of the center of $ U$. Using earlier results of the authors we show that this is a division algebra if and only if the group $ G$ is one of a very explicit (and short) list of nilpotent groups. It follows that for groups not on this list, one can find a nonidentity graded polynomial such that its power is a graded identity. We illustrate this phenomenon with an explicit example.


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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

Darrell Haile
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: haile@indiana.edu

Michael Natapov
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Address at time of publication: Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel
Email: mnatapov@indiana.edu, natapov@tx.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9947-10-04811-7
Received by editor(s): October 26, 2007
Received by editor(s) in revised form: April 22, 2008
Published electronically: January 7, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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