A NOTE ON GRADED GELFAND-KIRILLOV DIMENSION OF GRADED ALGEBRAS

In this paper, we consider associative P. I. algebras over a field F of characteristic 0, graded by a finite group G. More precisely, we define the G-graded Gelfand-Kirillov dimension of a G-graded P. I. algebra. We find a basis of the relatively free graded algebras of the upper triangular matrices UT(n)(F) and UT(n)(E), with entries in F and in the infinite-dimensional Grassmann algebra, respectively. As a consequence, we compute their graded Gelfand-Kirillov dimension with respect to the natural gradings defined over these algebras. We obtain similar results for the upper triangular matrix algebra UT(a,b)(E) = UT(a+b)(E)boolean AND M(a,b)(E) with respect to its natural Z(a+b) x Z(2)-grading. Finally, we compute the Z(n)-graded Gelfand-Kirillov dimension of M(n)(F) in some particular cases and with different methods.