Abstract
Given a finitely generated restricted Lie algebra L over the finite field \( \mathbb{F}_q \), and n ≥ 0, denote by a n (L) the number of restricted subalgebras H ⊆ L with \( \dim _{\mathbb{F} _q} \) L/H = n. Denote by ã 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 ≥ 2, we find the asymptotics of ã 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 ⊂ F d of finite codimension are maximal, which is analogous to the corresponding results for free groups and free associative algebras.
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Original Russian Text Copyright © 2008 Petrogradskiĭ V. M. and Smirnov A. A.
The authors were partially supported by the Russian Foundation for Basic Research (Grant 07-01-00080).
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Ul’yanovsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 6, pp. 1381–1390, November–December, 2008.
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Petrogradskiĭ, V.M., Smirnov, A.A. Enumeration of maximal subalgebras in free restricted lie algebras. Sib Math J 49, 1101–1108 (2008). https://doi.org/10.1007/s11202-008-0106-9
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DOI: https://doi.org/10.1007/s11202-008-0106-9