Cite this article as:

Петроградский В. М., Субботин И. А. About generating set of the invariant subalgebra of free restricted Lie algebra . Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2013, vol. 13, iss. 4, pp. 93-98. DOI: https://doi.org/10.18500/1816-9791-2013-13-4-93-98

Language: 
Russian
Heading: 
Mathematics
UDC: 
501.1

About generating set of the invariant subalgebra of free restricted Lie algebra

Authors: 
Petrogradsky Victor Mikhaylovich, University of Brasilia, Brasilia, Brazil, Brazil, Brasilia, Prédio da Reitoria , petrogradsky@rambler.ru
Subbotin Ivan Andreevich, Ulyanovsk State University, 42, Lev Tolstoy Str., Ulyanovsk, Russia, 432017 , shelby888@yandex.ru
Abstract: 

Suppose that L=L(X) is the free Lie p-algebra of finite rank k with free generating set X={x1,…,xk} on a field of positive characteristic. Let G is nontrivial finite group of homogeneous automorphisms L(X). Our main purpose to prove that LG subalgebra of invariants is is infinitely generated. We have more strongly result. Let Y=∪∞n=1Yn be homogeneous free generating set for the algebra of invariants LG, elements Yn are of degree n relatively X, n≥1. Consider the corresponding generating function H(Y,t)=∑∞n=1|Yn|tn. In our case of free Lie restricted algebras, we prove, that series H(Y,t) has a radius of convergence 1/k and describe its growth at t→1/k−0. As a result we obtain that the sequence |Yn|, n≥1, has exponential growth.

Key words: 
free Lie algebras, Lie p-algebras, invariants, generating set.
DOI: 
https://doi.org/10.18500/1816-9791-2013-13-4-93-98
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