A Note on Graded Gelfand–kirillov Dimension of Graded Algebras

Abstract

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 UTn(F) and UTn(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 UTa, b(E) = UTa+b(E)∩Ma, b(E) with respect to its natural ℤa+b × ℤ2-grading. Finally, we compute the ℤn-graded Gelfand–Kirillov dimension of Mn(F) in some particular cases and with different methods.

The graded Gelfand--Kirillov dimension of verbally prime algebras

“…In [5] we can also find an upper bound for the graded Gelfand-Kirillov dimensions of the verbally prime algebras.…”

“…Proof By Proposition 5.8 of [5], it follows that GKdim Z n ÂZ 2 k ðM a,b ðE ÞÞ kða 2 þ b 2 Þ À a þ 1:…”

“…, x g r k . We have the following theorem [5]: We find the Z n -graded Gelfand-Kirillov dimension for M n (F ) using a graded version of Balaba [2], of the well-known theorem of Posner (see, e.g. the book by Drensky and Formanek [10]).…”

“…We argue as in [5,Theorem 5.1] and we compute the graded Gelfand-Kirillov dimension of M 1,1 (E ) using standard combinatorial arguments. Let us consider the following definition.…”

“…Centrone [5] generalizes the notion of the Gelfand-Kirillov dimension for the algebras endowed with a G-grading, where G is a finite abelian group. In particular, he considers the relatively free G-graded algebra in k variables and defines the G-graded Gelfand-Kirillov dimension in k variables of a G-graded PI-algebra.…”