Integrality of exponents of codimension growth of finite-dimensional Lie algebras

We study the asymptotic behaviour of the codimension growth sequence $c_n(L)$ of a finite-dimensional Lie algebra $L$ over a field of characteristic zero. It is known that the growth of the sequence $\{c_n(L)\}$ is bounded by an exponential function of $n$, and hence there exist the upper and lower limits of the $n$th roots of $c_n(L)$, which are called the upper and lower exponents. By Amitsur's conjecture, the upper and lower exponents should coincide and be integers. This conjecture has been confirmed in the associative case for any PI-algebra. For finite-dimensional Lie algebras, a positive solution has been found for soluble, simple and semisimple algebras and also for algebras whose soluble radical is nilpotent. For infinite-dimensional Lie algebras, the problem has been solved in the negative. In this paper we give a proof of Amitsur's conjecture for arbitrary finite-dimensional Lie algebras.