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Supertraces and matrices over Grassmann algebras. (English) Zbl 0819.16023

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)

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)

Citations:

Zbl 0331.15021
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