×

Combinatorics on binary words and codimensions of identities in left nilpotent algebras. (English. Russian original) Zbl 1471.17025

Algebra Logic 58, No. 1, 23-35 (2019); translation from Algebra Logika 58, No. 1, 35-51 (2019).
Summary: Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent. And its precise value is computed.

MSC:

17B30 Solvable, nilpotent (super)algebras
05E16 Combinatorial aspects of groups and algebras
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Y. Bahturin and V. Drensky, Graded polynomial identities of matrices, Lin. Alg. Appl., 357, Nos. 1-3, 15-34 (2002). · Zbl 1019.16011
[2] A. Regev, “Existence of identities in <Emphasis Type=”Italic“>A ⊗ <Emphasis Type=”Italic“>B,” Isr. J. Math., 11, 131-152 (1972). · Zbl 0249.16007 · doi:10.1007/BF02762615
[3] S. P. Mishchenko, “Growth in varieties of Lie algebras,” Usp. Mat. Nauk, 45, No. 6 (276), 25-45 (1990). · Zbl 0718.17004
[4] M. V. Zaicev, “Identities of affine Katz-Moody algebras,” Vest. Mosk. Univ., Mat., Mekh., No. 2, 33-36 (1996). · Zbl 0881.17021
[5] M. V. Zaitsev, “Varieties of affine Katz-Moody algebras,” Mat. Zametki, 92, No. 1, 95-102 (1997). · doi:10.4213/mzm1591
[6] S. P. Mishchenko and V. M. Petrogradsky, “Exponents of varieties of Lie algebras with a nilpotent commutator subalgebra,” Comm. Alg., 27, No. 5, 2223-2230 (1999). · Zbl 0955.17002 · doi:10.1080/00927879908826560
[7] A. Giambruno and M. Zaicev, “On codimension growth of finitely generated associative algebras,” Adv. Math., 140, No. 2, 145-155 (1998). · Zbl 0920.16012 · doi:10.1006/aima.1998.1766
[8] A. Giambruno and M. Zaicev, “Exponential codimension growth of PI-algebras: An exact estimate,” Adv. Math., 142, No. 2, 221-243 (1999). · Zbl 0920.16013 · doi:10.1006/aima.1998.1790
[9] M. V. Zaicev, “Integrality of exponents of codimension growth of finite-dimensional Lie algebras,” Izv. Ross. Akad. Nauk, Mat., 66, No. 3, 23-48 (2002). · Zbl 1057.17003 · doi:10.4213/im386
[10] A. Giambruno, I. Shestakov, and M. Zaicev, “Finite-dimensional non-associative algebras and codimension growth,” Adv. Appl. Math., 47, No. 1, 125-139 (2011). · Zbl 1290.17028 · doi:10.1016/j.aam.2010.04.007
[11] M. V. Zaitsev and S. P. Mishchenko, “Identities for Lie superalgebras with a nilpotent commutator subalgebra,” Algebra and Logic, 47, No. 5, 348-364 (2008). · Zbl 1178.17005 · doi:10.1007/s10469-008-9022-0
[12] S. Mishchenko and M. Zaicev, “An example of a variety of Lie algebras with a fractional exponent”, J. Math. Sci., New York, 93, No. 6, 977-982 (1999). · Zbl 0933.17004 · doi:10.1007/BF02366352
[13] A. B. Verevkin, M. V. Zaitsev, and S. P. Mishchenko, “A sufficient condition for coincidence of lower and upper exponents of the variety of linear algebras,” Vest. Mosk. Univ., Ser. 1, Mat., Mekh., No. 2, 36-39 (2011). · Zbl 1304.17006
[14] A. Giambruno and M. Zaicev, “On codimension growth of finite-dimensional Lie superalgebras,” J. London Math. Soc., II. Ser., 85, No. 2, 534-548 (2012). · Zbl 1271.17003 · doi:10.1112/jlms/jdr059
[15] O. Malyusheva, S. Mishchenko, and A. Verevkin, “Series of varieties of Lie algebras of different fractional exponents,” C. R. Acad. Bulg. Sci., 66, No. 3, 321-330 (2013). · Zbl 1313.17011
[16] M. Zaicev, “On existence of PI-exponents of codimension growth,” El. Res. Announc. Math. Sci., 21, 113-119 (2014). · Zbl 1312.17005
[17] A. Giambruno, S. Mishchenko, and M. Zaicev, “Codimensions of algebras and growth functions,” Adv. Math., 217, No. 3, 1027-1052 (2008). · Zbl 1133.17001 · doi:10.1016/j.aim.2007.07.008
[18] M. V. Zaitsev, “Codimension growth of metabelian algebras,” Vest. Mosk. Univ., Ser. 1, Mat., Mekh., No. 6, 15-20 (2017).
[19] Yu. A. Bakhturin, Identities in Lie Algebras [in Russian], Nauka, Moscow (1985). · Zbl 0571.17001
[20] V. Drensky, Free Algebras and PI-Algebras. Graduate Course in Algebra, Springer, Singapore (2000). · Zbl 0936.16001
[21] A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Math. Surv. Monogr., 122, Am. Math. Soc., Providence, RI (2005). · Zbl 1105.16001
[22] G. James, The Representation Theory of the Symmetric Groups, Lect. Notes Math., 682, Springer (1978). · Zbl 0393.20009
[23] M. Lothaire, Algebraic Combinatorics on Words, Encycl. Math. Appl., 90, Cambridge Univ. Press, Cambridge (2002). · Zbl 1001.68093 · doi:10.1017/CBO9781107326019
[24] A. M. Shur, “Calculating parameters and behavior types of combinatorial complexity for regular languages,” Tr. Inst. Mat. Mech. UO RAN, 16, No. 2, 270-287 (2010).
[25] J. Balogh and B. Bollobás, “Hereditary properties of words,” Theor. Inform. Appl., 39, No. 1, 49-65 (2005). · Zbl 1132.68048 · doi:10.1051/ita:2005003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.