Abstract
We construct an additive basis of the free algebra of the variety generated by the model algebra of multiplicity 2 over an infinite field of characteristic not 2 and 3. Using the basis we remove a restriction on the characteristic in the theorem on identities of the model algebra (previously the same was proved in the case of characteristic 0). In particular, we prove that the kernel of the relatively free Lie-nilpotent algebra of index 5 coincides with the ideal of identities of the model algebra of multiplicity 2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Latyshev V. N., “On the choice of the basis in a T-ideal,” Sib. Mat. Zh., vol. 4, No. 5, 1122–1127 (1963).
Latyshev V. N., “Finite generation of a T-ideal with the element [x1, x2, x3, x4],” Sib. Mat. Zh., vol. 6, No. 6, 1432–1434 (1965).
Krakowski D. and Regev A., “The polynomial identities of the Grassmann algebra,” Trans. Amer. Math. Soc., vol. 181, 429–438 (1973).
Volichenko I. B., T-Ideal Generated by the Element [x1, x2, x3, x4] [Russian] [Preprint, no. 22], Inst. Mat. Akad. Nauk BSSR, Minsk (1978).
Grishin A. V., “An infinitely based T-ideal over a field of characteristic 2,” in: Abstracts: The International Conference on Algebra and Analysis Dedicated to N. G. Chebotarev on the Occasion of His 100 Birthday, Kazan, 5–11 June 1994, Kazan, 1994, 29.
Shchigolev V. V., “Examples of infinitely based T-ideals,” Fundam. Prikl. Mat., vol. 5, No. 1, 307–312 (1999).
Shchigolev V. V., “Examples of T-spaces with an infinite basis,” Sb. Math., vol. 191, No. 3, 459–476 (2000).
Grishin A. V., Tsybulya L. M., and Shokola A. A., “On T-spaces and relations in relatively free, Lie nilpotent, associative algebras,” J. Math. Sci., vol. 177, No. 6, 868–877 (2011).
Grishin A. V. and Pchelintsev S. V., “On centers of relatively free associative algebras with a Lie nilpotency identity,” Sb. Math., vol. 206, No. 11, 1610–1627 (2015).
Grishin A. V. and Pchelintsev S. V., “Proper central and core polynomials of relatively free associative algebras with identity of Lie nilpotency of degrees 5 and 6,” Sb. Math., vol. 207, No. 12, 674–1692 (2016).
Pchelintsev S. V., “Relatively free associative algebras of ranks 2 and 3 with Lie nilpotency identity and systems of generators of some T-spaces,” arXiv:1801.07771.
Pchelintsev S. V. and Shestakov I. P., “Constants of partial derivations and primitive operations,” Algebra and Logic, vol. 56, No. 3, 210–231 (2017).
Zhevlakov K. A., Slinko A. M., Shestakov I. P., and Shirshov A. I., Rings That Are Nearly Associative, Academic Press, New York (1982).
Jacobson N., Lie Algebras, Wiley and Sons, New York etc. (1962).
Buchnev A. A., Filippov V. T., and Shestakov I. P., “Checking identities of nonassociative algebras by computer,” in: III Siberian Congress on Applied and Industrial Mathematics (IN-PRIM-98), Izdat. Inst. Mat., Novosibirsk, 1998, 9.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text© 2018 Pchelintsev S.V.
__________
Moscow; Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, vol. 59, no. 6, pp. 1389–1411, November–December, 2018; DOI: 10.17377/smzh.2018.59.614.
Rights and permissions
About this article
Cite this article
Pchelintsev, S.V. Identities of the Model Algebra of Multiplicity 2. Sib Math J 59, 1105–1124 (2018). https://doi.org/10.1134/S0037446618060149
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446618060149