Abstract
In this note we exhibit bases of the polynomial identities satisfied by the Grassmann algebras over a field of positive characteristic. This allows us to answer the following question of Kemer: Does the infinite dimensional Grassmann algebra with 1, over an infinite fieldK of characteristic 3, satisfy all identities of the algebraM 2(K) of all 2×2 matrices overK? We give a negative answer to this question. Further, we show that certain finite dimensional Grassmann algebras do give a positive answer to Kemer's question. All this allows us to obtain some information about the identities satisfied by the algebraM 2(K) over an infinite fieldK of positive odd characteristic, and to conjecture bases of theidentities ofM 2(K).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Ya. Belov,On non-Spechtian varieties, Fundamental and Applied Mathematics5, No. 1 (1999), 47–66 (Russian).
A. Berele,Examples and counterexamples for M 1,1 embeddings, Journal of Algebra172, No. 2 (1995), 379–384.
P. Zh. Chiripov and P. N. Siderov,On bases for identities of some varieties of associative algebras, Pliska Studia Mathematica Bulgarica2 (1981), 103–115 (Russian).
O. M. Di Vincenzo,A note on the identities of the Grassmann algebras, Unione Matematica Italiana. Bollettino A7 (5), No. 3 (1991), 307–315.
O. M. Di Vincenzo and V. Drensky,Polynomial identities for tensor products of Grassmann algebras, Mathematica Pannonica4, No. 2 (1993), 249–272.
V. Drensky,A minimal basis of identities for a second-order matrix algebra over a field of characteristic 0, Algebra and Logic20, No. 3 (1981), 188–194.
V. Drensky,Identities of representations of nilpotent Lie algebras, Communications in Algebra25, No. 7 (1997), 2115–2127.
A. V. Grishin,Examples of T-spaces and T-ideals over a field of characteristic 2 without the finite basis property, Fundamental and Applied Mathematics5, No. 1 (1999), 101–118 (Russian).
A. R. Kemer,Ideals of identities of associative algebras, Translations of Mathematical Monographs 87, American Mathematical Society, Providence, RI, 1991.
A. Kemer,Identities of algebras over a field of characteristic p, Rendiconti del Circolo Matematico di Palermo. Supplemento31 (1993), 133–148.
A. R. Kemer,The standard identity in characteristic p. A conjecture of I. B. Volichenko, Israel Journal of Mathematics81 (1993), 343–355.
A. Kemer,Remarks on the prime varieties, Israel Journal of Mathematics96 (1995), 341–356.
A. Kemer,Multilinear identities of the algebras over a field of characteristic p, International Journal of Algebra and Computation5, No. 2 (1995), 189–197.
P. Koshlukov,Ideals of identities of representations of nilpotent Lie algebras, Communications in Algebra28 (2000), 3095–3113.
D. Krakowski and A. Regev,The polynomial identities of the Grassmann algebra, Transactions of the American Mathematical Society181 (1973), 429–438.
V. N. Latyshev,On the choice of basis in a T-ideal, Sibirskii Matematicheskii Zhurnal4, No. 5 (1963), 1122–1126 (in Russian).
A. Popov,On the identities of the matrices over the Grassmann algebra, Journal of Algebra168, No. 3 (1994), 828–852.
Yu. P. Razmyslov,Finite basing of the identities of a matrix algebra of second order over a field of characteristic zero, Algebra i Logika12, No. 1 (1973), 83–113 (Russian); English translation: Algebra and Logic12 (1973), 47–63.
A. Regev,Grassmann algebras over finite fields, Communications in Algebra19(6) (1991), 1829–1849.
V. V. Shchigolev,Examples of infinitely based T-ideals, Fundamental and Applied Mathematics5, No. 1 (1999), 307–312 (Russian).
W. Specht,Gesetze in Ringen. I, Mathematische Zeitschrift52 (1950), 557–589.
A. H. Stojanova-Venkova,Bases of identities of Grassmann algebras, Serdica6, No. 1 (1980), 63–72 (Russian).
S. Yu. Vasilovskii,The basis of identities of a three-dimensional simple Lie algebra over an infinite field, Algebra i Logika28, No. 5 (1989), 534–554 (Russian); English translation: Algebra and Logic28, No. 5 (1989), 355–368.
Author information
Authors and Affiliations
Corresponding author
Additional information
Partially supported by FAPESP and MURST.
Partially supported by FAPESP Grant 1998/0962-5.
Rights and permissions
About this article
Cite this article
Giambruno, A., Koshlukov, P. On the identities of the Grassmann algebras in characteristicp>0. Isr. J. Math. 122, 305–316 (2001). https://doi.org/10.1007/BF02809905
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02809905