Z-graded polynomial identities of the Grassmann algebra

2021 ◽  
Vol 617 ◽  
pp. 190-214
Author(s):  
Alan de Araújo Guimarães ◽  
Plamen Koshlukov
Keyword(s):  
Grassmann Algebra ◽  
Polynomial Identities ◽  
Graded Polynomial Identities

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).

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.

scholarly journals On Z2-graded polynomial identities of the Grassmann algebra

2009 ◽  
Vol 431 (1-2) ◽  
pp. 56-72 ◽  
Author(s):  
Onofrio M. Di Vincenzo ◽  
Viviane Ribeiro Tomaz da Silva
Keyword(s):  
Grassmann Algebra ◽  
Polynomial Identities ◽  
Graded Polynomial Identities

On the factorability of the ideal of ⁎-graded polynomial identities of minimal varieties of PI ⁎-superalgebras

2022 ◽  
Vol 589 ◽  
pp. 273-286
Author(s):  
Onofrio Mario Di Vincenzo ◽  
Viviane Ribeiro Tomaz da Silva ◽  
Ernesto Spinelli
Keyword(s):  
Polynomial Identities ◽  
Graded Polynomial Identities ◽  
The Ideal

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].

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