Codimension Growth of Lie algebras with a generalized action

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 H n (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. limn→∞ c H n (L) ∈ 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.