Graded identities of matrix algebras and the universal graded algebra Export

@article{citeulike:5657583, 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 <a href="/abs/0710.5468v1">arXiv:0710.5468v1</a> [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.}, archivePrefix = {arXiv}, author = {Aljadeff, Eli and Haile, Darrell and Natapov, Michael}, citeulike-article-id = {5657583}, citeulike-linkout-0 = {http://arxiv.org/abs/0710.5568}, citeulike-linkout-1 = {http://arxiv.org/pdf/0710.5568}, day = {30}, eprint = {0710.5568}, keywords = {algebras, identities, matrix, of}, month = {Oct}, posted-at = {2009-08-27 15:25:07}, priority = {2}, title = {Graded identities of matrix algebras and the universal graded algebra}, url = {http://arxiv.org/abs/0710.5568}, year = {2007} }

TY - JOUR ID - citeulike:5657583 L3 - citeulike-article-id:5657583 N2 - 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 <a href="/abs/0710.5468v1">arXiv:0710.5468v1</a> [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. TI - Graded identities of matrix algebras and the universal graded algebra KW - algebras KW - identities KW - matrix KW - of AU - Aljadeff, Eli AU - Haile, Darrell AU - Natapov, Michael PY - 2007/Oct/30/ UR - http://arxiv.org/abs/0710.5568 ER -