On Codimension Growth of Graded Pi-Algebras

Two finite dimensional simple Lie algebras over an algebraically closed field are isomorphic if and only if they satisfy the same polynomial identities (A. Koshkulei, Y. Razmyslov, 1983). As an alternative approach to the characterization of finite dimensional simple Lie algebras, some numerical invariants of the algebra identities can be used. We associate with a finite dimensional Lie algebra L a sequence of integers cₙ(L), called the n-th codimensions of L. It appears that these quantities grow asymptotically like kⁿ, for some nonnegative integer k≤ dim L (A. Giambruno, M. Zaicev, 1999). Moreover, $k =$ dim L iff algebra L is simple. * This paper was presented at the International Scientific Conference Graded structures in algebra and their applications, dedicated to the memory of Prof. Marc Krasner, IUCDubrovnik, Croatia, September, 22-24, 2016.