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Mat. Sb., 1997, Volume 188, Number 6, Pages 119–138 (Mi msb232)  

This article is cited in 36 scientific papers (total in 36 papers)

Growth of polynilpotent varieties of Lie algebras and rapidly growing entire functions

V. M. Petrogradsky

Ulyanovsk State University

Abstract: We study the growth function $c_n(\mathbf V)$ for a variety of Lie algebras, where $c_n(\mathbf V)$ is the dimension of the linear hull of the set of multilinear words with $n$ different letters in the free algebra $F(\mathbf V,X)$ of the variety $\mathbf V$. With every non-trivial variety $\mathbf V$ of Lie algebras there is associated its complexity function $\mathscr C(\mathbf V,z)$, which is an entire function of a complex variable. In the case of a polynilpotent variety $\mathbf V$ of Lie algebras an estimate is obtained for the complexity function; in most cases it is of infinite order. We study the connection between the growth of a rapidly growing entire function and the asymptotics of its Taylor coefficients. The basic result is the asymptotics for the function $c_n(\mathbf V)$ in the case of a polynilpotent variety $\mathbf V$. Also, we prove an analogue of Regev's theorem for Lie algebras on upper estimates for the growth of arbitrary varieties. This gives more precision to the scale of superexponential growth of varieties of Lie algebras introduced earlier by the author.

DOI: https://doi.org/10.4213/sm232

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English version:
Sbornik: Mathematics, 1997, 188:6, 913–931

Bibliographic databases:

UDC: 512.55
MSC: 17B99, 30D20
Received: 20.06.1996

Citation: V. M. Petrogradsky, “Growth of polynilpotent varieties of Lie algebras and rapidly growing entire functions”, Mat. Sb., 188:6 (1997), 119–138; Sb. Math., 188:6 (1997), 913–931

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    This publication is cited in the following articles:
    1. M. V. Zaicev, S. P. Mishchenko, “Asymptotic behaviour of the colength growth functions of varieties of Lie algebras”, Russian Math. Surveys, 54:3 (1999), 660–661  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Petrogradsky, VM, “Exponential generating functions and complexity of Lie varieties”, Israel Journal of Mathematics, 113 (1999), 323  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    3. V. M. Petrogradsky, “On numerical characteristics of subvarieties for three varieties of Lie algebras”, Sb. Math., 190:6 (1999), 887–902  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Petrogradsky, VM, “Growth of finitely generated polynilpotent Lie algebras and groups, generalized partitions, and functions analytic in the unit circle”, International Journal of Algebra and Computation, 9:2 (1999), 179  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    5. Mishchenko, SP, “Exponents of varieties of Lie algebras with a nilpotent commutator subalgebra”, Communications in Algebra, 27:5 (1999), 2223  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    6. V. M. Petrogradsky, “On the Complexity Functions for $T$-Ideals of Associative Algebras”, Math. Notes, 68:6 (2000), 751–759  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. Petrogradsky, VM, “On growth of Lie algebras, generalized partitions, and analytic functions”, Discrete Mathematics, 217:1–3 (2000), 337  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    8. Giambruno, A, “Simple and semisimple Lie algebras and codimension growth”, Transactions of the American Mathematical Society, 352:4 (2000), 1935  crossref  mathscinet  zmath  isi  elib
    9. M. V. Zaicev, “Integrality of exponents of codimension growth of finite-dimensional Lie algebras”, Izv. Math., 66:3 (2002), 463–487  mathnet  crossref  crossref  mathscinet  zmath  elib
    10. Petrogradsky, VM, “On generating functions for subalgebras of free Lie superalgebras”, Discrete Mathematics, 246:1–3 (2002), 269  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    11. Drensky V., “Polynomial identity rings - Part A - Combinatorial aspects in PI-rings”, Polynomial Identity Rings, Advanced Courses in Mathematics Crm Barcelona, 2004, 1  mathscinet  zmath  isi
    12. Petrogradsky, VM, “Enumeration of algebras close to absolutely free algebras and binary trees”, Journal of Algebra, 290:2 (2005), 337  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    13. Giambruno, A, “Algebras with intermediate growth of the codimensions”, Advances in Applied Mathematics, 37:3 (2006), 360  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    14. S. M. Ratseev, “The Growth of Varieties of Leibniz Algebras with Nilpotent Commutator Subalgebra”, Math. Notes, 82:1 (2007), 96–103  mathnet  crossref  crossref  mathscinet  isi  elib
    15. Giambruno, A, “Codimension growth of two-dimensional non-associative algebras”, Proceedings of the American Mathematical Society, 135:11 (2007), 3405  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    16. M. V. Zaitsev, S. P. Mishchenko, “Identities for Lie superalgebras with a nilpotent commutator subalgebra”, Algebra and Logic, 47:5 (2008), 348–364  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    17. S. S. Mishchenko, “On growth of varieties of commutative linear algebras”, J. Math. Sci., 163:6 (2009), 739–742  mathnet  crossref  mathscinet  elib  elib
    18. Giambruno, A, “Codimensions of algebras and growth functions”, Advances in Mathematics, 217:3 (2008), 1027  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    19. Mishchenko S., Valenti A., “On the growth of varieties of algebras”, Groups, Rings and Group Rings, Contemporary Mathematics, 499, 2009, 229–243  crossref  mathscinet  zmath  isi
    20. Giambruno A., Zaicev M., “Codimension Growth of Special Simple Jordan Algebras”, Transactions of the American Mathematical Society, 362:6 (2010), 3107–3123  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    21. Mishchenko S., Valenti A., “Varieties with at most quadratic growth”, Israel Journal of Mathematics, 178:1 (2010), 209–228  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    22. V. M. Petrogradsky, “Codimension Growth of Strong Lie Nilpotent Associative Algebras”, Comm. in Algebra, 39:3 (2011), 918  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    23. Di Vincenzo O.M., Nardozza V., “On the Existence of the Graded Exponent for Finite Dimensional Z(P)-Graded Algebras”, Can. Math. Bul.-Bul. Can. Math., 55:2 (2012), 271–284  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    24. ALI REZA SALEMKAR, ZAHRA RIYAHI, “THE POLYNILPOTENT MULTIPLIER OF Lie ALGEBRAS”, J. Algebra Appl, 12:02 (2013), 1250154  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    25. Sergey M. Ratseev, “On varieties of Leibniz–Poisson algebras with the identity $\{x,y\}\cdot ż,t\}=0$”, Zhurn. SFU. Ser. Matem. i fiz., 6:1 (2013), 97–104  mathnet
    26. Belov-Kanel A., Giambruno A., Rowen L.H., Vishne U., “Zariski Closed Algebras in Varieties of Universal Algebra”, Algebr. Represent. Theory, 17:6 (2014), 1771–1783  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    27. Mishchenko S., Valenti A., “An Almost Nilpotent Variety of Exponent 2”, Isr. J. Math., 199:1 (2014), 241–257  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    28. S. M. Ratseev, “On minimal Leibniz algebras with nilpotent commutator subalgebra”, St. Petersburg Math. J., 27:1 (2016), 125–136  mathnet  crossref  mathscinet  isi  elib
    29. S. M. Ratseev, “Complexity Functions of Varieties of Leibniz Algebras with Nilpotent Commutator Subalgebra”, Math. Notes, 98:3 (2015), 525–528  mathnet  crossref  crossref  mathscinet  isi  elib
    30. S. M. Ratseev, O. I. Cherevatenko, “Funktsii slozhnosti nekotorykh algebr Leibnitsa–Puassona”, Sib. elektron. matem. izv., 12 (2015), 500–507  mathnet  crossref
    31. S. M. Ratseev, O. I. Cherevatenko, “Korazmernosti mnogoobrazii algebr Puassona s lievo nilpotentnymi kommutantami”, Tr. IMM UrO RAN, 22, no. 1, 2016, 241–244  mathnet  mathscinet  elib
    32. S. M. Ratseev, “Numerical characteristics of varieties of Poisson algebras”, J. Math. Sci., 237:2 (2019), 304–322  mathnet  crossref
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    34. Mishchenko S. Valenti A., “Varieties With At Most Cubic Growth”, J. Algebra, 518 (2019), 321–342  crossref  mathscinet  zmath  isi  scopus
    35. Aslizadeh A., Salemkar A.R., Riyahi Z., “Polynilpotent Capability of Lie Algebras”, Commun. Algebr., 47:4 (2019), 1390–1400  crossref  isi
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