Abstract
We study metabelian alternative (in particular, associative) algebras over a field of characteristic 0. We construct additive bases of the free algebras of mentioned varieties, describe some centers of these algebras, compute the values of the sequence of codimensions of corresponding T-ideals, and find unitarily irreducible components of the decomposition of mentioned varieties into a union and their bases of identities. In particular, we find a basis of identities for the metabelian alternative Grassmann algebra. We prove that the free algebra of a variety that is generated by the metabelian alternative Grassmann algebra possesses the zero associative center.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Pchelintsev S. V., “Nilpotent elements and nil-radicals of alternative algebras,” Algebra and Logic, vol. 24, no. 6, 441–454 (1985).
Pchelintsev S. V., “Exceptional prime alternative algebras,” Sib. Math. J., vol. 48, no. 6, 1060–1073 (2007).
Pchelintsev S. V., “On torsion of a free alternative ring,” Sib. Math. J., vol. 32, no. 6, 1017–1023 (1991).
Pchelintsev S. V., “An almost Spechtian variety of alternative algebras over a field of characteristic 3,” Sb. Math., vol. 191, no. 6, 909–925 (2000).
Badeev A. V., “On the Specht property of varieties of commutative alternative algebras over a field of characteristic 3 and commutative Moufang loops,” Sib. Math. J., vol. 41, no. 6, 1027–1041 (2000).
Grishin A. V., “Examples of T-spaces and T-ideals over a field of characteristic 2 without the finite basis property,” Fundam. Prikl. Mat., vol. 5, 101–118 (1999).
Belov A. Ya., “Counterexamples to the Specht problem,” Sb. Math., vol. 191, no. 3, 329–340 (2000).
Shchigolev V. V., “Examples of T-spaces with an infinite basis,” Sb. Math., vol. 191, no. 3, 459–476 (2000).
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 [x 1, x 2, x 3, x 4],” 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).
Pchelintsev S. V., “A free (-1, 1)-algebra with two generators,” Algebra and Logic, vol. 13, no. 4, 241–257 (1974).
Volichenko I. B., T-Ideal Generated by the Element [x 1, x 2, x 3, x 4] [Preprint, no. 22], Inst. Mat. AN BSSR, Minsk (1978).
Bokut L. A. and Makar-Limanov L., “A basis of a free metabelian associative algebra,” Sib. Math. J., vol. 32, no. 6, 910–915 (1991).
Dorofeev G. V., “Varieties of generalized standard and generalized accessible algebras,” Algebra and Logic, vol. 15, no. 2, 90–104 (1976).
Dorofeev G. V. and Pchelintsev S. V., “The varieties of standard and accessible algebras,” Sib. Math. J., vol. 18, no. 5, 702–707 (1977).
Shestakov I. P., “Alternative algebras with identity [x, y]m = 0,” Algebra and Logic, vol. 20, no. 5, 377–392 (1981).
Pchelintsev S. V., “On the variety of algebras of type (-1, 1),” Algebra and Logic, vol. 25, no. 2, 97–108 (1986).
Pchelintsev S. V., “The structure of weak identities on the Grassmann envelopes of central-metabelian alternative superalgebras of superrank 1 over a field of characteristic 3,” Fundam. Prikl. Mat., vol. 7, no. 3, 849–871 (2001).
Shestakov I. P., “Free Malcev superalgebra on one odd generator,” J. Algebra Appl., vol. 2, no. 4, 451–461 (2003).
Shestakov I. P. and Zhukavets N., “The free alternative superalgebra on one odd generator,” Int. J. Algebra Comput., vol. 17, no. 5/6, 1215–1247 (2007).
Pchelintsev S. V., “On identities of right alternative metabelian Grassmann algebras,” J. Math. Sci. (New York), vol. 154, no. 2, 230–248 (2008).
Baulin A. N., Varieties of Alternative Algebras with Identity [x 1, x 2,..., x 5] = 0 [Russian], Avtoref. Diss. Kand. Fiz.-Mat. Nauk, Moscow (2005).
Pchelintsev S. V. and Shestakov I. P., “Prime (-1, 1) and Jordan monsters and superalgebras vector type,” J. Algebra, vol. 423, 54–86 (2015).
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).
Jacobson N., Structure and Representations of Jordan Algebras, Amer. Math. Soc., Providence (1968) (Amer. Math. Soc. Colloq. Publ.; vol. 39).
Zhevlakov K. A., Slin'ko A. M., Shestakov I. P., and Shirshov A. I., Rings That Are Nearly Associative, Academic Press, New York (1982).
Pchelintsev S. V., “Speciality of metabelian Mal'tsev algebras,” Math. Notes, vol. 74, no. 2, 245–254 (2003).
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).
Bakhturin Yu. A., Identities in Lie Algebras, VNU Science Press, Utrecht (1987).
Pchelintsev S. V. and Shestakov I. P., “Constants of partial derivations and primitive operations,” Algebra and Logic (2017) (to be published).
Grishin A. V., “On the structure of the centre of a relatively free Grassmann algebra,” Russian Math. Surveys, vol. 65, no. 4, 781–782 (2010).
Ryser H. J., Combinatorial Mathematics, Wiley, New York (1963).
Author information
Authors and Affiliations
Corresponding author
Additional information
Moscow; Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 58, No. 4, pp. 894–915, July–August, 2017
Rights and permissions
About this article
Cite this article
Pchelintsev, S.V. Identities of metabelian alternative algebras. Sib Math J 58, 693–710 (2017). https://doi.org/10.1134/S0037446617040164
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446617040164