Codimension growth of Lie algebras with a generalized action

Abstract

Let F F be a field of characteristic 0 0 and L L a finite dimensional Lie F F -algebra endowed with a generalized action by an associative algebra H H . We investigate the exponential growth rate of the sequence of H H -graded codimensions c n H ( L ) c_n^H(L) of L L which is a measure for the number of non-polynomial H H -identities of L L . More precisely, we construct an S S -graded Lie algebra (with S 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 → ∞ c n H ( L ) n ∈ Z \lim _{n\rightarrow \infty } \sqrt [n]{c_n^{H}(L)} \in \mathbb {Z} ) for general H H under the assumption that L L is both semisimple as Lie algebra and for the H H -action. Moreover if H = F S H=FS is a semigroup algebra the condition that L L is semisimple for the H 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.