On generating functions for subalgebras of free Lie superalgebras

  • Authors:

  • V. M. Petrogradsky

  • Affiliations:

  • Faculty of Mathematics, Ulyanovsk State University, Lev Tolstoy 42, 432700, Ulyanovsk, Russia

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2002

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Abstract

Let G be a free group of rank n and let H ∪ G be a subgroup of a finite index. Then H is also a free group and the rank m of H is determined by Schreier's formula m - 1 =(n - 1). |G : H|.Any subalgebra of a free Lie algebra is free (Shirshov-Witt). But a straightforward analogue of Schreier's formula for free Lie algebras does not exist, it is easy to see that any subalgebra of a finite codimension has an infinite number of generators.But the appropriate formula exists in terms of formal power series. The result is obtained in generality of free Lie superalgebras, graded by some semigroups. In this case instead of formal power series we use elements of the completion of a semigroup ring, which are called characters. As an application we specify the characters for free solvable Lie algebras. This result is established in generality of free polynilpotent Lie superalgebras.We also present similar results for exponential generating functions.