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Transactions of the American Mathematical Society Series B

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The Grassmann algebra over arbitrary rings and minus sign in arbitrary characteristic


Authors: Gal Dor, Alexei Kanel-Belov and Uzi Vishne
Journal: Trans. Amer. Math. Soc. Ser. B 7 (2020), 227-253
MSC (2010): Primary 16R10; Secondary 17A70, 16R30, 16R50
DOI: https://doi.org/10.1090/btran/49
Published electronically: November 18, 2020
MathSciNet review: 4175803
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Abstract:

An analog in characteristic $2$ for the Grassmann algebra $G$ was essential in a counterexample to the long standing Specht conjecture. We define a generalization $\mathfrak {G}$ of the Grassmann algebra, which is well-behaved over arbitrary commutative rings $C$, even when $2$ is not invertible. This lays the foundation for a supertheory over arbitrary base ring, allowing one to consider general deformations of superalgebras.

The construction is based on a generalized sign function. It enables us 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 $n$th co-module is a free $C$-module of rank $2^{n-1}$.


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Additional Information

Gal Dor
Affiliation: Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
MR Author ID: 985331
Email: dorgal111@gmail.com

Alexei Kanel-Belov
Affiliation: Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel; Moscow Institute of Physics and Technology, Institutskiy Pereulok, 9, Dolgoprudny, Moscow Oblast, Russia 141701
MR Author ID: 251623
ORCID: 0000-0002-1371-7479
Email: beloval@cs.biu.ac.il

Uzi Vishne
Affiliation: Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
MR Author ID: 626198
ORCID: 0000-0003-2760-9775
Email: vishne@math.biu.ac.il

Keywords: Superalgebra, generalized Grassmann algebra, generalized sign, polynomial identities, trace identities
Received by editor(s): July 9, 2015
Received by editor(s) in revised form: February 27, 2018, and August 7, 2019
Published electronically: November 18, 2020
Additional Notes: This work was supported by BSF grant 2010/149, ISF grants 1207/12 and 1994/20, and RSF grant 17-11-01377
Article copyright: © Copyright 2020 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)