Berele, Allan Supertraces and matrices over Grassmann algebras. (English) Zbl 0819.16023 Adv. Math. 108, No. 1, 77-90 (1994). Let \(M_ n(E)\) be the \(n \times n\) matrix algebra with entries from the Grassmann (or exterior) algebra over a field \(F\) of characteristic 0. The \(T\)-ideal of the polynomial identities for \(M_ n(E)\) is one of the building blocks of all \(T\)-ideals. The purpose of the paper under review is to establish superalgebra analogues of the results of C. Procesi [Adv. Math. 19, 306-381 (1976; Zbl 0331.15021)] who applied the classical invariant theory of the general linear group \(\text{GL}(n)\) to study trace identities for the ordinary \(n \times n\) matrix algebra \(M_ n(F)\).The author defines supertrace functions, constructs different supertraces for \(M_ n (E)\) and in the case of any of these supertraces gives generic models for \(M_ n(E)\) as a PI-algebra, as a graded PI-algebra and as an algebra with supertrace. The main results are that these generic supertrace algebras are the algebras of invariants of \(\text{GL}(n)\) and the general linear superalgebra \(\text{PL}(k,l)\) acting on a certain free supercommutative algebra. Finally the author generalizes the results to algebras with supertraces and superinvolution. Reviewer: V.Drensky (Sofia) Cited in 11 Documents MSC: 16R30 Trace rings and invariant theory (associative rings and algebras) 15A75 Exterior algebra, Grassmann algebras 16W55 “Super” (or “skew”) structure 15A72 Vector and tensor algebra, theory of invariants 16R10 \(T\)-ideals, identities, varieties of associative rings and algebras 17A70 Superalgebras 16R50 Other kinds of identities (generalized polynomial, rational, involution) Keywords:matrices over Grassmann algebra; exterior algebra; matrix algebra; polynomial identities; \(T\)-ideals; invariant theory of the general linear group; trace identities; supertraces; PI-algebra; graded PI; generic supertrace algebras; algebras of invariants; general linear superalgebra; superinvolution Citations:Zbl 0331.15021 PDFBibTeX XMLCite \textit{A. Berele}, Adv. Math. 108, No. 1, 77--90 (1994; Zbl 0819.16023) Full Text: DOI