Graded identities of matrix algebras and the universal graded algebra

TL;DR

This paper studies fine G-gradings on complex matrix algebras, characterizes their graded identities, constructs universal graded algebras, and explores their properties and division algebra conditions.

Contribution

It introduces elementary binomial identities generating the graded identities and constructs universal algebras via polynomial identities and cocycles.

Findings

Graded identities are finitely generated by elementary binomials.

Universal algebra U_{G,c} is Azumaya over its center.

Certain graded quotient algebras are division algebras for specific groups.

Abstract

We consider fine G-gradings on M_n(C) (i.e. gradings of the matrix algebra over the complex numbers where each component is 1 dimensional). Groups which provide such a grading are known to be solvable. We consider the T-ideal of G-graded identities and show that it is generated by a special type of binomial identities which we call elementary. In particular we show that the ideal of graded identities is finitely generated as a T-ideal. Next, given such grading we construct a universal algebra U_{G,c} 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 non-graded case). We show that a suitable central localization of U_{G,c} is Azumaya over its center and moreover, its homomorphic images are precisely the G-graded forms of M_n(C). Finally, we consider the ring of central quotients Q(U_{G,c}) (this is an F-central simple algebra where F=Frac(Z) and Z is the center of of U_{G,c}). Using an earlier results of the authors (see E. Aljadeff, D. Haile and M. Natapov, Projective bases of division algebras and groups of central type, Israel J. Math.146 (2005) 317-335 and M. Natapov arXiv:0710.5468v1 [math.RA]) we show that this is a division algebra for a very explicit (and short) family of nilpotent groups. As a consequence, for groups G such that Q(U_{G,c}) is not a division algebra, one can find a non identity polynomial p(x_{i,g}) such that p(x_{i,g})^r is a graded identity for some integer r. We illustrate this phenomenon with a fine G-grading of M_6(C) where G is a semidirect product of S_3 and C_6.