ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

7. Lie Groups and Lie Algebras

1991 MS Classification: 16, 17 Petrogradsky Victor, Ulyanovsk State University, Russia On subalgebras of free Lie Algebras Let X be a set with the weight function wt : X → N such that there exists the series H(X, t) = ∑_{i=1}^{∞} |X_{i}|t^{i}, X_{i} = {x ∈ X| wt x = i}. Let L be the free Lie algebra generated by X. All series for sets and spaces are defined with respect to this weight function in the conventional way. Let E be the operator on power series E(∑_{n=1}^{∞} b_{n}t^{n}) = ∏_{n=1}^{∞}(1 − t^{n})^{−b_{n}}. The following is the analogue of Schreier’s formula for free groups. Theorem. Suppose that H ⊂ L is a subalgebra with a free generating set Y. Then H(Y, t) − 1 = (H(X, t) − 1) · E(H(L/H, t)). There is a version for exponential generating functions. Their apllications will be presented.