Superpolynomial identities of finite-dimensional simple algebras

2024/06/25 by Bahturin, Yuri, Yasumura, Felipe Yukihide
#FOS: Mathematics #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.2406.17270

Abstract

We investigate the Grassmann envelope (of finite rank) of a finite-dimensional ℤ₂-graded algebra. As a result, we describe the polynomial identities of G₁(A), where G₁ stands for the Grassmann algebra with 1 generator, and A is a ℤ₂-graded-simple associative algebra. We also classify the conditions under which two associative ℤ₂-graded-simple algebras share the same set of superpolynomial identities, i.e., the polynomial identities of its Grassmann envelope (in particular, of finite rank). Moreover, we extend the construction of the Grassmann envelope for the context of Ω-algebras and prove some of its properties. Lastly, we give a description of ℤ₂-graded-simple Ω-algebras.

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• ℤ-gradings of full support on the Grassmann algebra
• ℤ₂-graded identities of the Grassmann algebra over a finite field (update version)
• Automorphisms and superalgebra structures on the Grassmann algebra
• ℤ-graded polynomial identities of the Grassmann algebra