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

Author(s)

Fidelis, Claudemir ; Guimaraes, Alan ; Koshlukov, Plamen

• Document type: Journal article
• Source: LINEAR & MULTILINEAR ALGEBRA; v. N/A, p. 21-pg., 2022-04-07.

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 Z-gradings on E having the form E =E-(r1,r2,r3)((v1,v2,v3)) in which each element of a basis of L has 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 and we describe the graded identities for the cor responding 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 Brandao A, Fidelis C, Guimaraes A. Z-gradings of full support on the Grassmann algebra. (AU)

• FAPESP's process: 18/23690-6 - Structures, representations, and applications of algebraic systems
• Grantee: Ivan Chestakov
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 19/12498-0 - Graded polynomial identities and identity with trace, and invariant theory
• Grantee: Claudemir Fideles Bezerra Júnior
• Support Opportunities: Scholarships in Brazil - Post-Doctoral