Automorphisms and superalgebra structures on the Grassmann algebra

Author(s)

Guimaraes, Alan de Araujo ; Koshlukov, Plamen

• Document type: Journal article
• Source: Journal of Pure and Applied Algebra; v. 227, n. 1, p. 20-pg., 2023-01-01.

Abstract

Let F be a field of characteristic zero and let E be the Grassmann algebra of an infinite dimensional F-vector space L. In this paper we study the superalgebra structures (that is the Z(2)-gradings) that the algebra E admits. By using the duality between superalgebras and automorphisms of order at most 2 we prove that in many cases the Z(2)-graded polynomial identities for such structures coincide with the Z(2)-graded polynomial identities of the "typical " cases E-infinity, E-k* and E(k )where the vector space L is homogeneous. Recall that these cases were completely described by Di Vincenzo and da Silva in [13]. Moreover we exhibit a wide range of nonhomogeneous Z(2)-gradings on E that are Z(2)-isomorphic to E-infinity, E-k* and E-k. In particular we construct a Z(2)-grading on E with only one homogeneous generator in L which is Z(2)-isomorphic to the natural Z(2)-grading on E, here denoted by E-can. (c) 2022 Elsevier B.V. All rights reserved. (AU)

• FAPESP's process: 18/23690-6 - Structures, representations, and applications of algebraic systems
• Grantee: Ivan Chestakov
• Support Opportunities: Research Projects - Thematic Grants