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

This article is cited in 37 scientific papers (total in 37 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.


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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

Citation in format AMSBIB
\by V.~M.~Petrogradsky
\paper Growth of polynilpotent varieties of Lie algebras and rapidly growing entire functions
\jour Mat. Sb.
\yr 1997
\vol 188
\issue 6
\pages 119--138
\jour Sb. Math.
\yr 1997
\vol 188
\issue 6
\pages 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
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    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
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    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
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    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
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    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
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    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
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    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
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    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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  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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