A note on -gradings on the Grassmann algebra and elementary number theory

Abstract

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 $\mathbb{Z}$-gradings on E having the form $E=E_{(r_1,r_2,r_3)}^{(v_1,v_2,v_3)}$, in which each element of a basis of L has $\mathbb{Z}$-degree $r_1,r_2$, or $r_3$. We provide a criterion for the support of this structure to coincide with a subgroup of the group $\mathbb{Z}$, 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 Brandão A, Fidelis C, Guimarães A. $\mathbb{Z}$-gradings of full support on the Grassmann algebra. J Algebra. 2022;601:332–353. DOI:10.1016/j.jalgebra.2022.03.014. See also in arXiv preprint, arXiv:2009.01870v1, 2020].

COMMUNICATED BY:

• M. Chebotar

Keywords:

• Grassmann algebra
• graded algebra
• graded identity
• full support
• greatest common divisor

Mathematics Subject Classification 2010:

• 11A05
• 11A51
• 15A75
• 16R10
• 16W50

Acknowledgments

The authors thank the Referee for her/his the valuable comments and suggestions that improved the presentation of this paper. We are particularly grateful for the Referee's suggestion to improve Theorem 3.4.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Claudemir Fidelis was supported by São Paulo Research Foundation (FAPESP) [grant number 2019/12498-0]. Alan Guimarães was partially supported by CNPq [grant number 404851/2021-5] and by FAPESQ-PB [grant number 2021/3176]. Plamen Koshlukov was partially supported by São Paulo Research Foundation (FAPESP) [grant number 2018/23690-6] and by CNPq [grant number 302238/2019-0].