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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.

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) |