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Petrogradsky V. M., Subbotin I. A. About generating set of the invariant subalgebra of free restricted Lie algebra. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2013, vol. 13, iss. 4, pp. 93-98. DOI: 10.18500/1816-9791-2013-13-4-93-98

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

Petrogradsky Victor Mikhaylovich, University of Brasilia
Subbotin Ivan Andreevich, Ulyanovsk State University

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.

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