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Superpolynomial identities of finite-dimensional simple algebras

Yuri Bahturin, Felipe Yukihide Yasumura

Abstract

We investigate the Grassmann envelope (of finite rank) of a finite-dimensional $\mathbb{Z}_2$-graded algebra. As a result, we describe the polynomial identities of $G_1(\mathcal{A})$, where $G_1$ stands for the Grassmann algebra with $1$ generator, and $\mathcal{A}$ is a $\mathbb{Z}_2$-graded-simple associative algebra. We also classify the conditions under which two associative $\mathbb{Z}_2$-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 $\mathbb{Z}_2$-graded-simple $Ω$-algebras.

Superpolynomial identities of finite-dimensional simple algebras

Abstract

We investigate the Grassmann envelope (of finite rank) of a finite-dimensional -graded algebra. As a result, we describe the polynomial identities of , where stands for the Grassmann algebra with generator, and is a -graded-simple associative algebra. We also classify the conditions under which two associative -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 -algebras and prove some of its properties. Lastly, we give a description of -graded-simple -algebras.

Paper Structure

This paper contains 14 sections, 28 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Main results: Associative case
  4. Structure results on $G_m(\mathcal{A})$
  5. Polynomial identities of $G_1(\mathcal{A})$
  6. Distinguishing simple algebras by their superpolynomial identities
  7. $\Omega$-algebras
  8. Preliminaries
  9. Gradings on $\Omega$-algebras
  10. Codimension sequence
  11. Grassmann envelope
  12. Polynomial identities of the Grassmann algebra
  13. Graded-simple algebras
  14. Some generalization

Key Result

Proposition 1

Let $\mathcal{A}=\mathcal{A}^0\oplus\mathcal{A}^1$ be a finite-dimensional associative $\mathbb{Z}_2$-graded algebra. Assume that $\mathcal{A}^0=\mathcal{S}+J$, where $\mathcal{S}$ is a semisimple subalgebra, $J=J(\mathcal{A}^0)$ is the Jacobson radical of $\mathcal{A}^0$, and $\mathcal{S}\cap J=0$. and $J+G_m^\ast(\mathcal{A})$ is the Jacobson radical of $G_m(\mathcal{A})$.

Theorems & Definitions (52)

  • Proposition 1
  • proof
  • Corollary 2
  • proof
  • Corollary 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • proof
  • ...and 42 more