Exponential Schreier's formula for free Lie algebras and its applications
The author studies the growth function \(c_n({\mathbb V})\) for a variety of Lie algebras \({\mathbb V}\), where \(c_n({\mathbb V})\) is the dimension of the linear hull of the set of multilinear words with \(n\) different letters in the free algebra of the variety \({\mathbb V}\). For every non-trivial variety of linear algebras \({\mathbb V}\), some complexity function \({\mathcal C}({\mathbb V},z)\) is constructed which is an entire function of a complex variable. Namely, \({\mathcal C}({\mathbb V},z)=\sum_{n=1}^{\infty} c_n({\mathbb V})z^n/n!\), \(z\) is a complex variable. The author obtains the exponential analogue of Schreier's formula for free groups (Theorem 2.2). Further he applies this result to the determination of explicit formulas for the complexity function of the product of Lie varieties (Theorem 3.1) and that of the polynomial variety (Theorem 3.2). Finally, the author finds asymptotics for some concrete varieties (Theorems 4.3 and 4.4).
scientific article; zbMATH DE number 1315228
scientific article