## 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)

Zbl 0331.15021
Full Text: