The graded Gelfand--Kirillov dimension of verbally prime algebras

Abstract

Let E be the infinite-dimensional Grassmann algebra over a field F of characteristic 0. In this article, we consider the verbally prime algebras M n (F ), M n (E ) and M a,b (E ) endowed with their gradings induced by that of Vasilovsky, and we compute their graded Gelfand-Kirillov dimensions.

ℤ2-Graded Gelfand–Kirillov dimension of the Grassmann algebra

“…In all of the cases we have considered we have studied the growth function of F Y, Z /T 2 (A), where |Y | = |Z| = m. This restriction was reasonable in the papers [6,5] by the author because if A is a superalgebra, then the sub-algebra of F Y, Z /T 2 (A) generated by x i = y i + z i has the same identities of the algebra F X /T (A) and is easier to be studied. One may consider also the case when Y and Z have different number of elements.…”

“…We refer to the surveys [10] of Drensky and [7] of the author and the papers [6,5] by the author for more details. Notice that we have defined the G-graded GK dimension of A in k variables as the GK dimension of F G k (A) that is generated by sk variables.…”

Z_2-graded Gelfand-Kirillov dimension of the Grassmann algebra

“…m (E ∞ ) that is merely the GK dimension of a relatively-free algebra of 2m variables "modulo" the identity [u 1 , u 2 , u 3 ]. It turns out it must be equal to the GK dimension of E in 2m variables as we may verify by [6,4].…”

On the growth of graded polynomial identities of

“…, x gs k ) ∩ T G (L) the relatively-free G-graded algebra of L in k variables. We define the G-graded Gelfand-Kirillov dimension of L in k variables similarly to the associative case (see [9,10]),…”

“…In particular if A is a G-graded PI algebra, one may consider the relatively-free G-graded algebra of A in k variables F G k (A) and define the G-graded Gelfand-Kirillov dimension of A as the GK dimension of F G k (A). In [10] Centrone computed the graded Gelfand-Kirillov dimension of the verbally prime algebras M n (K), M n (E) and M a,b (E), where K is a field and E is the infinite dimensional Grassmann algebra, endowed with a "Vasilovsky type"-grading (see [26]). We recall that the verbally prime algebras are the building blocks of the theory of Kemer.…”