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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
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 Γ of finite dimensional algebras Σ with Id (Σ) = Γ. By Kemer’s theory Γ is not empty. We show there exists A Γ with Wedderburn–Malcev decomposition AAss JA, where JA is the Jacobson’s radical and Ass is a semisimple supplement with the property that if BBss JB Γ then Ass is a direct summand of Bss. In particular Ass is unique minimal, thus an invariant of Γ. More generally, let Γ be the T-ideal of identities of a PI algebra and let 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 2,Γ is not empty. We prove there exists a superalgebra AAss JA 2,Γ such that if B 2,Γ, then Ass is a direct summand of Bss 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 G2 := 2 × G-graded simple algebras then they are G2-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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