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Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 2013, Volume 13, Issue 4(2), Pages 93–98 (Mi isu476)  

Mathematics

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

V. M. Petrogradskya, I. A. Subbotinb

a Department of Mathematics, University of Brasilia, 70910-900 Brasilia DF, Brazil
b Ulyanovsk State University, Russia, 432970, Ulyanovsk, ul. L'va Tolstogo, 42

Abstract: Suppose that $L=L(X)$ is the free Lie p-algebra of finite rank $k$ with free generating set $X=\{x_1,…,x_k\}$ on a field of positive characteristic. Let $G$ is nontrivial finite group of homogeneous automorphisms $L(X)$. Our main purpose to prove that $L^G$ subalgebra of invariants is is infinitely generated. We have more strongly result. Let $Y=\cup_{n=1}^\infty Y_n$ be homogeneous free generating set for the algebra of invariants $L^G$, elements $Y_n$ are of degree $n$ relatively $X$, $n\ge1$. Consider the corresponding generating function $\mathscr H(Y,t)=\sum_{n=1}^\infty|Y_n|t^n$. In our case of free Lie restricted algebras, we prove, that series $\mathscr H(Y,t)$ has a radius of convergence $1/k$ and describe its growth at $t\to1/k-0$. As a result we obtain that the sequence $|Y_n|$, $n\ge1$, has exponential growth.

Key words: free Lie algebras, Lie p-algebras, invariants, generating set.

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Document Type: Article
UDC: 501.1

Citation: V. M. Petrogradsky, I. A. Subbotin, “About generating set of the invariant subalgebra of free restricted Lie algebra”, Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform., 13:4(2) (2013), 93–98

Citation in format AMSBIB
\Bibitem{PetSub13}
\by V.~M.~Petrogradsky, I.~A.~Subbotin
\paper About generating set of the invariant subalgebra of free restricted Lie algebra
\jour Izv. Saratov Univ. (N.S.), Ser. Math. Mech. Inform.
\yr 2013
\vol 13
\issue 4(2)
\pages 93--98
\mathnet{http://mi.mathnet.ru/isu476}


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  • Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
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