On the codimension growth of finite-dimensional Lie algebras
Let \(V\) be a variety of linear algebras and \(F(V)\) its countable rank relatively free algebra generated by elements \(y_1,y_2,\dots\). Let \(P_n(V)\subset F(V)\) be a subspace of multilinear elements in \(y_1,\dots,y_n\) in \(F(V)\). Denote \(c_n(V)=\roman{dim}P_n(V)\), often these numbers are referred to as the (codimension) growth sequence. For varieties of associative algebras the codimension growth is at most exponential (A. Regev). Moreover, by recent result of M. Zaitsev and A. Giambruno, over fields of zero characteristic there exists a limit \(\lim_{n\to\infty}\root n\of{c_n(V)}=k\), where \(k\) is an integer. For Lie algebras the situation is more complicated. It is known that the codimension growth might be overexponential. If \(L\) is an algebra, then \(c_n(V)\) is the growth of the variety generated by \(L\). As a main result the authors prove: Theorem: Let \(L\) be a finite-dimensional Lie algebra over a field of characteristic zero, whose solvable radical is nilpotent. Then there exists the limit \(\lim_{n\to\infty}\root n\of{c_n(L)}=k\), where \(k\) is an integer.
scientific article; zbMATH DE number 1384469