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

Dor, Gal; Kanel-Belov, Alexei; Vishne, Uzi

doi:10.1090/btran/49

The Grassmann algebra over arbitrary rings and minus sign in arbitrary characteristic
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by Gal Dor, Alexei Kanel-Belov and Uzi Vishne HTML | PDF

Trans. Amer. Math. Soc. Ser. B 7 (2020), 227-253

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 $2^{n-1}$.

References

Eli Aljadeff and Alexei Kanel-Belov, Representability and Specht problem for $G$-graded algebras, Adv. Math. 225 (2010), no. 5, 2391–2428. MR 2680170, DOI 10.1016/j.aim.2010.04.025
A. Ya. Belov, Counterexamples to the Specht problem, Mat. Sb. 191 (2000), no. 3, 13–24 (Russian, with Russian summary); English transl., Sb. Math. 191 (2000), no. 3-4, 329–340. MR 1773251, DOI 10.1070/SM2000v191n03ABEH000460
Alexei Kanel-Belov and Louis Halle Rowen, Computational aspects of polynomial identities, Research Notes in Mathematics, vol. 9, A K Peters, Ltd., Wellesley, MA, 2005. MR 2124127, DOI 10.1201/9781439863725
Pierre Deligne and John W. Morgan, Notes on supersymmetry (following Joseph Bernstein), Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) Amer. Math. Soc., Providence, RI, 1999, pp. 41–97. MR 1701597
Antonio Giambruno and Plamen Koshlukov, On the identities of the Grassmann algebras in characteristic $p>0$, Israel J. Math. 122 (2001), 305–316. MR 1826505, DOI 10.1007/BF02809905
A. V. Grishin and L. M. Tsybulya, On the $T$-space and multiplicative structure of a relatively free Grassmann algebra, Mat. Sb. 200 (2009), no. 9, 41–80 (Russian, with Russian summary); English transl., Sb. Math. 200 (2009), no. 9-10, 1299–1338. MR 2583970, DOI 10.1070/SM2009v200n09ABEH004038
A. V. Grishin, L. M. Tsybulya, and A. A. Shokola, On $T$-spaces and relations in relatively free Lie nilpotent associative algebras, Fundam. Prikl. Mat. 16 (2010), no. 3, 135–148 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 177 (2011), no. 6, 868–877. MR 2786534, DOI 10.1007/s10958-011-0515-3
Antonio Giambruno and Mikhail Zaicev, Polynomial identities and asymptotic methods, Mathematical Surveys and Monographs, vol. 122, American Mathematical Society, Providence, RI, 2005. MR 2176105, DOI 10.1090/surv/122
Aaron Kaufer, Superalgebra in characteristic $2$, 2018. arXiv:1804.00824.
Aleksandr Robertovich Kemer, Ideals of identities of associative algebras, Translations of Mathematical Monographs, vol. 87, American Mathematical Society, Providence, RI, 1991. Translated from the Russian by C. W. Kohls. MR 1108620, DOI 10.1090/mmono/087
Alexander Kemer, Multilinear identities of the algebras over a field of characteristic $p$, Internat. J. Algebra Comput. 5 (1995), no. 2, 189–197. MR 1328550, DOI 10.1142/S0218196795000124
D. Krakowski and A. Regev, The polynomial identities of the Grassmann algebra, Trans. Amer. Math. Soc. 181 (1973), 429–438. MR 325658, DOI 10.1090/S0002-9947-1973-0325658-5
Issai Kantor and Ivan Trishin, On a concept of determinant in the supercase, Comm. Algebra 22 (1994), no. 10, 3679–3739. MR 1280094, DOI 10.1080/00927879408825050
Joel Louswa, Adilson Eduardo Presto, and Alan Tarr, A new computation of the codimension sequence of the Grassmann algebra, Rose-Hulman Undergraduate Math. J. 6 (2005), no. 2.
L. M. Tsybulya, Theorems on equalization and monomiality in a relatively free Grassmann algebra, Fundam. Prikl. Mat. 14 (2008), no. 5, 197–218 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 163 (2009), no. 6, 759–773. MR 2533589, DOI 10.1007/s10958-009-9714-6
Siddharth Venkatesh, Hilbert basis theorem and finite generation of invariants in symmetric tensor categories in positive characteristic, Int. Math. Res. Not. IMRN 16 (2016), 5106–5133. MR 3556434, DOI 10.1093/imrn/rnv305
E. I. Zel′manov, Superalgebras and identities, Algebra and analysis (Kemerovo, 1988) Amer. Math. Soc. Transl. Ser. 2, vol. 148, Amer. Math. Soc., Providence, RI, 1991, pp. 39–46. MR 1109062, DOI 10.1090/trans2/148/04