Semisimple algebras and PI-invariants of finite dimensional algebras

Aljadeff, Eli; Karasik, Yakov

doi:10.2140/ant.2024.18.133

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Semisimple algebras and PI-invariants of finite dimensional algebras

Eli Aljadeff and Yakov Karasik

Vol. 18 (2024), No. 1, 133–164

DOI: 10.2140/ant.2024.18.133

Abstract

Let $Γ$ be the $T$ -ideal of identities of an affine PI-algebra over an algebraically closed field $F$ of characteristic zero. Consider the family $ℳ_{Γ}$ of finite dimensional algebras $Σ$ with $Id (Σ) = Γ$ . By Kemer’s theory $ℳ_{Γ}$ is not empty. We show there exists $A \in ℳ_{Γ}$ with Wedderburn–Malcev decomposition $A ≅ A_{s s} \oplus J_{A}$ , where $J_{A}$ is the Jacobson’s radical and $A_{s s}$ is a semisimple supplement with the property that if $B ≅ B_{s s} \oplus J_{B} \in ℳ_{Γ}$ then $A_{s s}$ is a direct summand of $B_{s s}$ . In particular $A_{s s}$ is unique minimal, thus an invariant of $Γ$ . More generally, let $Γ$ be the $T$ -ideal of identities of a PI algebra and let $ℳ_{ℤ_{2}, Γ}$ be the family of finite dimensional superalgebras $Σ$ with $Id (E (Σ)) = Γ$ . Here $E$ is the unital infinite dimensional Grassmann algebra and $E (Σ)$ is the Grassmann envelope of $Σ$ . Again, by Kemer’s theory $ℳ_{ℤ_{2}, Γ}$ is not empty. We prove there exists a superalgebra $A ≅ A_{s s} \oplus J_{A} \in ℳ_{ℤ_{2}, Γ}$ such that if $B \in ℳ_{ℤ_{2}, Γ}$ , then $A_{s s}$ is a direct summand of $B_{s s}$ as superalgebras. Finally, we fully extend these results to the $G$ -graded setting where $G$ is a finite group. In particular we show that if $A$ and $B$ are finite dimensional $G_{2} : = ℤ_{2} \times G$ -graded simple algebras then they are $G_{2}$ -graded isomorphic if and only if $E (A)$ and $E (B)$ are $G$ -graded PI-equivalent.

Keywords

T-ideal, polynomial identities, semisimple, full algebras, graded algebras

Mathematical Subject Classification

Primary: 16R10, 16W50, 16W55

Milestones

Received: 7 February 2022

Revised: 30 October 2022

Accepted: 7 December 2022

Published: 22 November 2023

Authors

Eli Aljadeff
Department of Mathematics Technion Haifa Israel

Yakov Karasik
Department of Mathematics Technion Haifa Israel

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Vol. 18, No. 1, 2024