Zq–graded identities and central polynomials of the Grassmann algebra

2021 ◽  
Vol 609 ◽  
pp. 12-36
Author(s):  
Alan Guimarães ◽  
Claudemir Fidelis ◽  
Laise Dias
Keyword(s):  
Grassmann Algebra ◽  
Graded Identities

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

2016 ◽  
Vol 26 (06) ◽  
pp. 1125-1140 ◽  
Author(s):  
Lucio Centrone ◽  
Viviane Ribeiro Tomaz da Silva
Keyword(s):  
Positive Characteristic ◽  
Finite Abelian Group ◽  
Grassmann Algebra ◽  
Graded Algebra ◽  
Infinite Field ◽  
Infinite Dimensional ◽  
Graded Identities ◽  
Graded Polynomial Identities ◽  
Specific Formula ◽  
Gk Dimension

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].

scholarly journals Graded identities for tensor products of matrix (super)algebras over the Grassmann algebra

2010 ◽  
Vol 432 (2-3) ◽  
pp. 780-795 ◽  
Author(s):  
Onofrio Mario Di Vincenzo ◽  
Plamen Koshlukov ◽  
Ednei Aparecido Santulo
Keyword(s):  
Tensor Products ◽  
Grassmann Algebra ◽  
Graded Identities

Graded identities for algebras with elementary gradings over an infinite field

2022 ◽  
pp. 1-21
Author(s):  
Diogo Diniz ◽  
Claudemir Fidelis ◽  
Plamen Koshlukov
Keyword(s):  
Tensor Product ◽  
Associative Algebra ◽  
Positive Characteristic ◽  
Grassmann Algebra ◽  
Alternative Proof ◽  
Infinite Field ◽  
Infinite Dimensional ◽  
Graded Identities ◽  
Graded Polynomial Identities ◽  
Field Of Positive Characteristic

Abstract Let $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$ -grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$ , $b\in \mathbb {N}$ , we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$ , as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$ -algebras which are not PI equivalent. Actually, we prove that the $T_{G}$ -ideal of the former algebra is contained in the $T$ -ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

On ℤp-graded identities and cocharacters of the Grassmann algebra

2016 ◽  
Vol 45 (1) ◽  
pp. 343-356 ◽  
Author(s):  
Onofrio Mario Di Vincenzo ◽  
Plamen Koshlukov ◽  
Viviane Ribeiro Tomaz Da Silva
Keyword(s):  
Grassmann Algebra ◽  
Graded Identities

ℤ2-graded identities of the Grassmann algebra over a finite field

2018 ◽  
Vol 28 (02) ◽  
pp. 291-307 ◽  
Author(s):  
Luís Felipe Gonçalves Fonseca
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Grassmann Algebra ◽  
Dimensional Vector ◽  
Polynomial Identities ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Graded Identities ◽  
Graded Polynomial Identities

Let [Formula: see text] be a finite field with the characteristic [Formula: see text] and let [Formula: see text] be the unitary Grassmann algebra generated by an infinite dimensional vector space [Formula: see text] over [Formula: see text]. In this paper, we determine a basis for [Formula: see text]-graded polynomial identities for any [Formula: see text]-grading such that its underlying vector space is homogeneous.

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