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 | References | Similar Articles | Additional Information

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)