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