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 Sibirsk. Mat. Zh., 2008, Volume 49, Number 6, Pages 1381–1390 (Mi smj1927)

Enumeration of maximal subalgebras in free restricted Lie algebras

V. M. Petrogradsky, A. A. Smirnov

Ulyanovsk State University, Faculty of Mathematics and Information Technologies

Abstract: Given a finitely generated restricted Lie algebra $L$ over the finite field $\mathbb F_q$, and $n\ge0$, denote by $a_n(L)$ the number of restricted subalgebras $H\subseteq L$ with $\dim_{\mathbb F_q}L/H=n$. Denote by $\widetilde a_n(L)$ the number of the subalgebras satisfying the maximality condition as well. Considering the free restricted Lie algebra $L=F_d$ of rank $d\ge2$, we find the asymptotics of $\widetilde a_n(F_d)$ and show that it coincides with the asymptotics of $a_n(F_d)$ which was found previously by the first author. Our approach is based on studying the actions of restricted algebras by derivations on the truncated polynomial rings. We establish that the maximal subalgebras correspond to the so-called primitive actions. This means that “almost all” restricted subalgebras $H\subset F_d$ of finite codimension are maximal, which is analogous to the corresponding results for free groups and free associative algebras.

Keywords: restricted Lie algebra, Witt algebra, coalgebras, enumerative combinatorics, subgroup growth.

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English version:
Siberian Mathematical Journal, 2008, 49:6, 1101–1108

Bibliographic databases:

UDC: 512.55

Citation: V. M. Petrogradsky, A. A. Smirnov, “Enumeration of maximal subalgebras in free restricted Lie algebras”, Sibirsk. Mat. Zh., 49:6 (2008), 1381–1390; Siberian Math. J., 49:6 (2008), 1101–1108

Citation in format AMSBIB
\Bibitem{PetSmi08} \by V.~M.~Petrogradsky, A.~A.~Smirnov \paper Enumeration of maximal subalgebras in free restricted Lie algebras \jour Sibirsk. Mat. Zh. \yr 2008 \vol 49 \issue 6 \pages 1381--1390 \mathnet{http://mi.mathnet.ru/smj1927} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2499108} \transl \jour Siberian Math. J. \yr 2008 \vol 49 \issue 6 \pages 1101--1108 \crossref{https://doi.org/10.1007/s11202-008-0106-9} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000261792400015} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-57749198261} 

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This publication is cited in the following articles:
1. V. M. Petrogradsky, I. A. Subbotin, “Ideal growth in metabelian Lie $p$-algebras”, Siberian Math. J., 56:4 (2015), 714–724
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