Exponential generating functions and complexity of Lie varieties
Let \({\mathfrak V}\) be a variety of Lie algebras, \(c_n({\mathfrak V})\) be the dimension of the linear span of all multilinear words with \(n\) distinct letters in the free algebra of the variety \({\mathfrak V}\). For every non-trivial variety of linear algebras \({\mathfrak V}\), some complexity function \({\mathcal C}({\mathfrak V},z)\) is constructed which is an entire function of a complex variable. Namely, \({\mathcal C}({\mathfrak V},z)=\sum_{n=1}^{\infty} c_n({\mathfrak V}) z^n/n!\), \(z\) is a complex variable. In the article under review, the author introduces the notion of complexity for varieties of Lie algebras. The main goal of the author is to specify the complexity of a product of two varieties in terms of the complexities of multiplicands. The main observation is that \({\mathcal C}({\mathfrak {MV}},z)\) behaves like a composition of functions \({\mathcal C}({\mathfrak M},z)\), \(\exp(z)\), and \({\mathcal C}({\mathfrak V},z)\).
scientific article; zbMATH DE number 1389714
scientific article