Codimensions of polynomial identities of representations of Lie algebras (Q2845413)

Amitsur's conjecture on the asymptotic behavior of the codimensions of polynomial identities was proved by \textit{A. Giambruno} and \textit{M. Zaicev} [Adv. Math. 142, No. 2, 221--243 (1999; Zbl 0920.16013)] for associative algebras and by \textit{M. V. Zaitsev} [Izv. Math. 66, No. 3, 463--487 (2002; Zbl 1057.17003); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 3, 23--48 (2002)] for finite dimensional Lie algebras. The present paper extends these results for identities of representations of Lie algebras. An element \(f\) of the free associative algebra \(F\langle X_1,\dots,X_n\rangle\) over a field \(F\) of characteristic zero is a polynomial identity of the finite dimensional representation \(\rho:L\to \mathfrak{gl}(V)\) if \(f(\rho(a_1),\dots,\rho(a_n))=0\) in \(\mathfrak{gl}(V)\) for all \(a_1,\dots,a_n\in L\). The \(n\)th codimension \(c_n(\rho)\) is the codimension of the space of multilinear polynomial identities of \(\rho\) in the space of all multilinear elements in \(F\langle X_1,\dots,X_n\rangle\). The main result of the paper (obtained as a corollary of a more precise statement) is that \(\lim_{n\to \infty}\root{n}\of{c_n(\rho)}\) exists and is a non-negative integer.