Semisimple algebras and PI-invariants of finite dimensional algebras

Abstract

Let Γ be a 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 it is known that such Σ exists.We show there exists a semisimple algebra U which satisfies the following conditions.(1) There exists an algebra A ∈ M Γ with Wedderburn-Malcev decomposition A ∼ = U ⊕ J A , where J A is the Jacobson's radical of A (2) If B ∈ M Γ and B ∼ = Bss ⊕ J B is its Wedderburn-Malcev decomposition then U is a direct summand of Bss.We refer to U as the unique minimal semisimple algebra corresponding to Γ.More generally, if Γ is a T -ideal of identities of a PI algebra and M Z 2 ,Γ is the family of finite dimensional super algebras 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 it is known that such Σ exists.Then there exists a semisimple super algebra U with the following properties.(1) There exists an algebra A ∈ M Z 2 ,Γ with Wedderburn-Malcev decomposition as super algebrasits Wedderburn-Malcev decomposition as super algebras, then U is a direct summand of Bss as super algebras.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 × 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.