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 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 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 nth co-module is a free C-module of rank 2n−1.
| Original language | English |
|---|---|
| Pages (from-to) | 227-253 |
| Number of pages | 27 |
| Journal | Transactions of the American Mathematical Society Series B |
| Volume | 7 |
| DOIs | |
| State | Published - 2020 |
Bibliographical note
Publisher Copyright:© 2020 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0).
Funding
Received by the editors July 9, 2015, and, in revised form, February 27, 2018, and August 7, 2019. 2010 Mathematics Subject Classification. Primary 16R10; Secondary 17A70, 16R30, 16R50. Key words and phrases. Superalgebra, generalized Grassmann algebra, generalized sign, polynomial identities, trace identities. This work was supported by BSF grant 2010/149, ISF grants 1207/12 and 1994/20, and RSF grant 17-11-01377.
| Funders | Funder number |
|---|---|
| United States-Israel Binational Science Foundation | 2010/149 |
| Israel Science Foundation | 1207/12, 1994/20 |
| Russian Science Foundation | 17-11-01377 |
Keywords
- Generalized Grassmann algebra
- Generalized sign
- Polynomial identities
- Superalgebra
- Trace identities