<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msub><mml:mrow><mml:mi mathvariant="double-struck">Z</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math>–graded identities and central polynomials of the Grassmann algebra

Guimarães, Alan; Fidelis, Claudemir; Dias, Laise

doi:10.1016/j.laa.2020.08.014

Cited by 5 publications

(3 citation statements)

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“…polynomials for E. In [14] the Z q -graded identities and central polynomials for all homogeneous Z q -gradings on E were described, for q > 2. In [15] the authors developed the first study related to non-homogeneous Z 2 -gradings on E. They proved that there exist infinitely many non-homogeneous such gradings on E.…”

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confidence: 99%

$\mathbb{Z}$-gradings of full support on the Grassmann algebra

Guimarães¹,

Brandão²,

Fidelis³

2020

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Let E be the infinite dimensional Grassmann algebra over a field F of characteristic zero. In this paper we investigate the structures of Zgradings on E of full support. Using methods of elementary number theory, we describe the Z-graded polynomial identities for the so-called 2-induced Z-gradings on E of full support. As a consequence of this fact we provide examples of Z-gradings on E which are PI-equivalent but not Z-isomorphic. This is the first example of graded algebras with infinite support that are PI-equivalent and not isomorphic as graded algebras. We also present the notion of central Z-gradings on E and we show that its Z-graded polynomial identities are closely related to the Z2-graded polynomial identities of Z2-gradings on E.

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confidence: 99%

$\mathbb{Z}$-gradings of full support on the Grassmann algebra

Guimarães¹,

Brandão²,

Fidelis³

2020

Preprint

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“…(Here and in what follows Z 2 stands for the cyclic group of order 2.) Furthermore, in [2,4,12], gradings and graded identities on E by finite abelian groups of order larger than 2 were also investigated. Gradings on E by infinite groups were studied for the first time in [13].…”

Section: Introductionmentioning

confidence: 99%

“…When the field K is infinite, one has to consider multihomogeneous identities, for more details of this technique we recommend [5,Section 4.2]. The methods one uses in this case are mostly based on Combinatorial Algebra [1,12,13]. A technique that has been used recently is based on Elementary Number Theory.…”

Section: Introductionmentioning

confidence: 99%

A note on $\mathbb{Z}$-gradings on the Grassmann algebra and Elementary Number Theory

Guimarães¹,

Fidelis²,

Koshlukov³

2021

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Let E be the Grassmann algebra of an infinite dimensional vector space L over a field of characteristic zero. In this paper, we study the Z-gradings on E having the form, in which each element of a basis of L has Z-degree r1, r2, or r3. We provide a criterion for the support of this structure to coincide with a subgroup of the group Z, and we describe the graded identities for the corresponding gradings. We strongly use Elementary Number Theory as a tool, providing an interesting connection between this classical part of Mathematics, and PI Theory. Our results are generalizations of the approach presented in [11].

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