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)


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)


Zbl 0331.15021
Full Text: DOI