A note on graded polynomial identities for tensor products by the Grassmann algebra in positive characteristic

Abstract

Let [Formula: see text] be a finite abelian group. As a consequence of the results of Di Vincenzo and Nardozza, we have that the generators of the [Formula: see text]-ideal of [Formula: see text]-graded identities of a [Formula: see text]-graded algebra in characteristic 0 and the generators of the [Formula: see text]-ideal of [Formula: see text]-graded identities of its tensor product by the infinite-dimensional Grassmann algebra [Formula: see text] endowed with the canonical grading have pairly the same degree. In this paper, we deal with [Formula: see text]-graded identities of [Formula: see text] over an infinite field of characteristic [Formula: see text], where [Formula: see text] is [Formula: see text] endowed with a specific [Formula: see text]-grading. We find identities of degree [Formula: see text] and [Formula: see text] while the maximal degree of a generator of the [Formula: see text]-graded identities of [Formula: see text] is [Formula: see text] if [Formula: see text]. Moreover, we find a basis of the [Formula: see text]-graded identities of [Formula: see text] and also a basis of multihomogeneous polynomials for the relatively free algebra. Finally, we compute the [Formula: see text]-graded Gelfand–Kirillov (GK) dimension of [Formula: see text].

On Z2-graded identities of UT2(

“…We observe that the same happens for the Grassmann algebra E in positive characteristic endowed with its canonical Z 2 -grading (see Theorem 7.1 of [6] for more details).…”

“…In this example we are focusing on θ 1 and we will work with the equation β 0 1 θ 1 +β 3 1 θ 4 + β 6 1 θ 7 = 0. Since F is an infinite field, then there exist distinct β 1,0 , β 1,1 , β 1,2 ∈ F − {0}.…”

On the graded identities of the Grassmann algebra

“…Let F be an infinite field of characteristic p > 2, then we have the next results (see [7]). Theorem 4.2.…”

“…In this section we collect the results concerning the Z 2 -graded identities of E based on the works by the author [7] and Di Vincenzo and da Silva [14]. For the sake of completeness we want to cite the papers [37] and [2] by Anisimov in which the author computes the sequence of involutive codimensions of Grassmann algebra for some special involutions (see [37]), then generalized in [2].…”

The $G$-graded identities of the Grassmann Algebra

“…Recently, Di Vincenzo and da Silva gave in [6] a complete description of the Z 2 -graded polynomial identities of E with respect to any Z 2 -grading such that the generating space is Z 2 -homogeneous. This work has been generalized by the author for any infinite field of characteristic p > 2 (see [2]). In [1] Anisimov constructed an algorithm to compute the exact value of the graded codimension of E for any Z p -grading of E, where p is a prime number.…”