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Izv. RAN. Ser. Mat., 2002, Volume 66, Issue 3, Pages 23–48 (Mi izv386)  

This article is cited in 51 scientific papers (total in 51 papers)

Integrality of exponents of codimension growth of finite-dimensional Lie algebras

M. V. Zaicev

M. V. Lomonosov Moscow State University

Abstract: We study the asymptotic behaviour of the codimension growth sequence $c_n(L)$ of a finite-dimensional Lie algebra $L$ over a field of characteristic zero. It is known that the growth of the sequence $\{c_n(L)\}$ is bounded by an exponential function of $n$, and hence there exist the upper and lower limits of the $n$th roots of $c_n(L)$, which are called the upper and lower exponents. By Amitsur's conjecture, the upper and lower exponents should coincide and be integers. This conjecture has been confirmed in the associative case for any PI-algebra. For finite-dimensional Lie algebras, a positive solution has been found for soluble, simple and semisimple algebras and also for algebras whose soluble radical is nilpotent. For infinite-dimensional Lie algebras, the problem has been solved in the negative. In this paper we give a proof of Amitsur's conjecture for arbitrary finite-dimensional Lie algebras.


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English version:
Izvestiya: Mathematics, 2002, 66:3, 463–487

Bibliographic databases:

UDC: 512.8
MSC: Primary 17B01, 17B20, 16R10; Secondary 20C30, 17C05
Received: 28.02.2001

Citation: M. V. Zaicev, “Integrality of exponents of codimension growth of finite-dimensional Lie algebras”, Izv. RAN. Ser. Mat., 66:3 (2002), 23–48; Izv. Math., 66:3 (2002), 463–487

Citation in format AMSBIB
\by M.~V.~Zaicev
\paper Integrality of exponents of codimension growth of finite-dimensional Lie algebras
\jour Izv. RAN. Ser. Mat.
\yr 2002
\vol 66
\issue 3
\pages 23--48
\jour Izv. Math.
\yr 2002
\vol 66
\issue 3
\pages 463--487

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    2. Giambruno A., Mishchenko S., Zaicev M., “Codimension growth of two-dimensional non-associative algebras”, Proc Amer Math Soc, 135:11 (2007), 3405–3415  crossref  mathscinet  zmath  isi  elib  scopus
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    4. 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
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  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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