On Z2-graded identities of UT2(

Abstract

Let F be an infinite field of characteristic different from two and E be the infinite dimensional Grassmann algebra over F . We consider the upper triangular matrix algebra UT 2 (E) with entries in E endowed with the Z 2 -grading inherited by the natural Z 2 -grading of E and we study its ideal of Z 2 -graded polynomial identities (T Z2 -ideal) and its relatively free algebra. In particular we show that the set of Z 2 -graded polynomial identities of UT 2 (E) does not depend on the characteristic of the field. Moreover we compute the Z 2 -graded Hilbert series of UT 2 (E) and its Z 2 -graded Gelfand-Kirillov dimension.

On the graded identities of the Grassmann algebra

“…Theorem 6.3. (C. and da Silva [9]) Let F be a field of characteristic p > 2 and E be graded with its natural Z 2 -grading. Let us set S := {[y 1 , y 2 ], [y 1 , z 1 ], z 1 z 2 • z 2 z 1 }, then T Z2 (U T 2 (E)) = {ζ J (f )|f ∈ S}.…”

Cocharacters of $UT_n(E)$

“…The next theorem talks about the polynomial identities of the F -algebra U T n (E) of n × n upper triangular matrices with entries in the Grassmann algebra E. See also [15] for the case of U T 2 (E) in positive characteristic.…”