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].
In the present paper it is proved that a variety of associative PI-superalgebras with graded involution of finite basic rank over a field of characteristic zero is minimal of fixed * -graded exponent if, and only if, it is generated by a subalgebra of an upper block triangular matrix algebra equipped with a suitable elementary Z2-grading and graded involution.
It has been shown that in characteristic zero the generators of the minimal supervarieties of finite basic rank belong to the class of minimal superalgebras introduced by Giambruno and Zaicev (Trans Am Math Soc 355:5091â\u80\u935117, 2003). In the present paper the complete list of minimal supervarieties generated by minimal superalgebras whose maximal semisimple homogeneous subalgebra is the sum of three graded simple algebras is provided. As a consequence, we negatively answer the question of whether any minimal superalgebra generates a minimal supervariety
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.
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