Codimension growth of Lie algebras with a generalized action

Abstract:

Let $F$ be a field of characteristic $0$ and $L$ a finite dimensional Lie $F$-algebra endowed with a generalized action by an associative algebra $H$. We investigate the exponential growth rate of the sequence of $H$-graded codimensions $c_n^H(L)$ of $L$ which is a measure for the number of non-polynomial $H$-identities of $L$. More precisely, we construct an $S$-graded Lie algebra (with $S$ a semigroup) which has an irrational exponential growth rate (the exact value is obtained). This is the first example of a graded Lie algebra with non-integer exponential growth rate. In addition, we prove an analogue of Amitsur’s conjecture (i.e. $\lim _{n\rightarrow \infty } \sqrt [n]{c_n^{H}(L)} \in \mathbb {Z}$) for general $H$ under the assumption that $L$ is both semisimple as Lie algebra and for the $H$-action. Moreover if $H=FS$ is a semigroup algebra the condition that $L$ is semisimple for the $H$-action can be dropped. This is in strong contrast to the associative setting where an infinite family of graded-simple algebras with irrational graded PI-exponent was constructed.