Algebra and Logic

, Volume 58, Issue 1, pp 23–35 | Cite as

Combinatorics on Binary Words and Codimensions of Identities in Left Nilpotent Algebras

  • M. V. ZaicevEmail author
  • D. D. Repovš

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.


left nilpotent algebra polynomial identity codimension subexponential function of combinatorial complexity PI-exponent 


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  1. 1.
    Y. Bahturin and V. Drensky, Graded polynomial identities of matrices, Lin. Alg. Appl., 357, Nos. 1-3, 15-34 (2002).Google Scholar
  2. 2.
    A. Regev, “Existence of identities in AB,” Isr. J. Math., 11, 131-152 (1972).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    S. P. Mishchenko, “Growth in varieties of Lie algebras,” Usp. Mat. Nauk, 45, No. 6 (276), 25-45 (1990).Google Scholar
  4. 4.
    M. V. Zaicev, “Identities of affine Katz–Moody algebras,” Vest. Mosk. Univ., Mat., Mekh., No. 2, 33-36 (1996).Google Scholar
  5. 5.
    M. V. Zaitsev, “Varieties of affine Katz–Moody algebras,” Mat. Zametki, 92, No. 1, 95-102 (1997).CrossRefGoogle Scholar
  6. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. Giambruno and M. Zaicev, “On codimension growth of finitely generated associative algebras,” Adv. Math., 140, No. 2, 145-155 (1998).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    A. Giambruno and M. Zaicev, “Exponential codimension growth of PI-algebras: An exact estimate,” Adv. Math., 142, No. 2, 221-243 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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).MathSciNetCrossRefGoogle Scholar
  10. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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).Google Scholar
  14. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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).MathSciNetzbMATHGoogle Scholar
  16. 16.
    M. Zaicev, “On existence of PI-exponents of codimension growth,” El. Res. Announc. Math. Sci., 21, 113-119 (2014).MathSciNetzbMATHGoogle Scholar
  17. 17.
    A. Giambruno, S. Mishchenko, and M. Zaicev, “Codimensions of algebras and growth functions,” Adv. Math., 217, No. 3, 1027-1052 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    M. V. Zaitsev, “Codimension growth of metabelian algebras,” Vest. Mosk. Univ., Ser. 1, Mat., Mekh., No. 6, 15-20 (2017).Google Scholar
  19. 19.
    Yu. A. Bakhturin, Identities in Lie Algebras [in Russian], Nauka, Moscow (1985).zbMATHGoogle Scholar
  20. 20.
    V. Drensky, Free Algebras and PI-Algebras. Graduate Course in Algebra, Springer, Singapore (2000).zbMATHGoogle Scholar
  21. 21.
    A. Giambruno and M. Zaicev, Polynomial Identities and Asymptotic Methods, Math. Surv. Monogr., 122, Am. Math. Soc., Providence, RI (2005).Google Scholar
  22. 22.
    G. James, The Representation Theory of the Symmetric Groups, Lect. Notes Math., 682, Springer (1978).Google Scholar
  23. 23.
    M. Lothaire, Algebraic Combinatorics on Words, Encycl. Math. Appl., 90, Cambridge Univ. Press, Cambridge (2002).CrossRefzbMATHGoogle Scholar
  24. 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).Google Scholar
  25. 25.
    J. Balogh and B. Bollobás, “Hereditary properties of words,” Theor. Inform. Appl., 39, No. 1, 49-65 (2005).MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia
  2. 2.Univerza v LjubljaniLjubljanaSlovenia

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