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arXiv · 1501.02464

The Grassmann algebra in arbitrary characteristic and generalized sign

Gal Dor·Alexei Kanel-Belov·Uzi Vishne
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Abstract

We define a generalization $\mathfrak{G}$ of the Grassmann algebra $G$ which is well-behaved over arbitrary commutative rings $C$, even when $2$ is not invertible. In particular, this enables us to define a notion of superalgebras that does not become degenerate in such a setting. Using this construction we are able to provide a basis of the non-graded multilinear identities of the free superalgebra with supertrace, valid over any ring. We also show that all identities of $\mathfrak{G}$ follow from the Grassmann identity, and explicitly give its co-modules, which turn out to be generalizations of the sign representation. In particular, we show that the co-module is a free $C$-module of rank $2^{n-1}$.

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Gal Dor, Alexei Kanel-Belov, Uzi Vishne. 2015-01-11. The Grassmann algebra in arbitrary characteristic and generalized sign. https://doi.org/10.1090/btran%2F49

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