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 of by matrices over the complex numbers and the corresponding graded polynomial identities. Given a group and a fine -grading on , we show that the -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 (depending on the group 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 is Azumaya over its center and moreover, its homomorphic images are precisely the -graded forms of . Finally, we consider the ring of central quotients of which is a central simple algebra over the field of quotients of the center of . Using earlier results of the authors we show that this is a division algebra if and only if the group 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.