TY - JOUR
T1 - Graded identities of matrix algebras and the universal graded algebra
AU - Aljadeff, Eli
AU - Haile, Darrell
AU - Natapov, Michael
PY - 2010/6
Y1 - 2010/6
N2 - We consider fine group gradings on the algebra Mn(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 Mn(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 Mn(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.
AB - We consider fine group gradings on the algebra Mn(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 Mn(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 Mn(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.
UR - http://www.scopus.com/inward/record.url?scp=77951639913&partnerID=8YFLogxK
U2 - 10.1090/S0002-9947-10-04811-7
DO - 10.1090/S0002-9947-10-04811-7
M3 - 文章
AN - SCOPUS:77951639913
SN - 0002-9947
VL - 362
SP - 3125
EP - 3147
JO - Transactions of the American Mathematical Society
JF - Transactions of the American Mathematical Society
IS - 6
ER -