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].

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.

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.

ℤ2-Graded Gelfand–Kirillov dimension of the Grassmann algebra

2014 ◽  
Vol 24 (03) ◽  
pp. 365-374 ◽  
Author(s):  
Lucio Centrone
Keyword(s):  
Grassmann Algebra ◽  
Infinite Dimensional ◽  
Gk Dimension ◽  
Kirillov Dimension

We consider the infinite dimensional Grassmann algebra E over a field F of characteristic 0 or p, where p > 2, and we compute its ℤ2-graded Gelfand–Kirillov (GK) dimension as a ℤ2-graded PI-algebra.

scholarly journals THE CENTRAL POLYNOMIALS OF THE INFINITE-DIMENSIONAL UNITARY AND NONUNITARY GRASSMANN ALGEBRAS

2010 ◽  
Vol 09 (05) ◽  
pp. 687-704 ◽  
Author(s):  
C. BEKH-OCHIR ◽  
S. A. RANKIN
Keyword(s):  
Grassmann Algebra ◽  
Infinite Field ◽  
Characteristic P ◽  
Infinite Dimensional ◽  
Grassmann Algebras ◽  
Unitary Algebra ◽  
Algebra Over A Field

We describe the T-space of central polynomials for both the unitary and the nonunitary infinite-dimensional Grassmann algebra over a field of characteristic p≠2 (infinite field in the case of the unitary algebra).

GRADED POLYNOMIAL IDENTITIES OF VERBALLY PRIME ALGEBRAS

2007 ◽  
Vol 06 (03) ◽  
pp. 385-401 ◽  
Author(s):  
ONOFRIO M. DI VINCENZO ◽  
VINCENZO NARDOZZA
Keyword(s):  
Tensor Product ◽  
Vector Space ◽  
Characteristic Zero ◽  
Grassmann Algebra ◽  
Polynomial Identities ◽  
Graded Algebras ◽  
Infinite Dimensional ◽  
Graded Polynomial Identities ◽  
New Type

Let F be a field and let E be the Grassmann algebra of an infinite dimensional F-vector space. For any p,q ∈ ℕ, the algebra Mp,q(E) can be turned into a ℤp+q × ℤ2-algebra by combining an elementary ℤp+q-grading with the natural ℤ2-grading on E. The tensor product Mp,q(E) ⊗ Mr,s(E) can be turned into a ℤ(p+q)(r+s) × ℤ2-algebra in a similar way. In this paper, we assume that F has characteristic zero and describe a system of generators for the graded polynomial identities of the algebras Mp,q(E) and Mp,q(E) ⊗ Mr,s(E) with respect to these new gradings. We show that this tensor product is graded PI-equivalent to Mpr+qs,ps+qr(E). This provides a new proof of the well known Kemer's PI-equivalence between these algebras. Then we classify all the graded algebras Mp,q(E) having no non-trivial monomial identities, and finally calculate how many non-isomorphic gradings of this new type are available for Mp,q(E).

GRADED IDENTITIES FOR THE ALGEBRA OF n×n UPPER TRIANGULAR MATRICES OVER AN INFINITE FIELD

2003 ◽  
Vol 13 (05) ◽  
pp. 517-526 ◽  
Author(s):  
PLAMEN KOSHLUKOV ◽  
ANGELA VALENTI
Keyword(s):  
Infinite Field ◽  
Polynomial Identities ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Graded Identities ◽  
Graded Polynomial Identities ◽  
The Ideal

We consider the algebra Un(K) of n×n upper triangular matrices over an infinite field K equipped with its usual ℤn-grading. We describe a basis of the ideal of the graded polynomial identities for this algebra.

scholarly journals GRADED TWISTED CALABI–YAU ALGEBRAS ARE GENERALIZED ARTIN–SCHELTER REGULAR

2021 ◽  
pp. 1-54
Author(s):  
MANUEL L. REYES ◽  
DANIEL ROGALSKI
Keyword(s):  
Regularity Property ◽  
Graded Algebra ◽  
General Study ◽  
Locally Finite ◽  
Tensor Algebras ◽  
Finite Dimensional ◽  
Dimension 2 ◽  
Separable Algebras ◽  
Gk Dimension

Abstract This is a general study of twisted Calabi–Yau algebras that are $\mathbb {N}$ -graded and locally finite-dimensional, with the following major results. We prove that a locally finite graded algebra is twisted Calabi–Yau if and only if it is separable modulo its graded radical and satisfies one of several suitable generalizations of the Artin–Schelter regularity property, adapted from the work of Martinez-Villa as well as Minamoto and Mori. We characterize twisted Calabi–Yau algebras of dimension 0 as separable k-algebras, and we similarly characterize graded twisted Calabi–Yau algebras of dimension 1 as tensor algebras of certain invertible bimodules over separable algebras. Finally, we prove that a graded twisted Calabi–Yau algebra of dimension 2 is noetherian if and only if it has finite GK dimension.

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