Growth of Lie algebras and the existence of the classical of fractions for the universal enveloping algebras
The purpose of the paper under review is to relate the growth of a Lie algebra \(L\) with the existence of a classical ring of fractions of its universal enveloping algebra \(U(L)\). The main result is the following. Let \({\mathcal V}\) be a variety of Lie algebras such that all finitely generated relatively free algebras in \({\mathcal V}\) are of subexponential growth. If \(L\) is a Lie algebra with a solvable ideal \(I\) such that \(L/I\in{\mathcal V}\), then the universal enveloping algebra \(U(L)\) has a classical ring of fractions. In particular, this result holds for the variety \({\mathcal V}\) generated by a finite dimensional algebra. In his approach the author uses the Ore condition and techniques typical for the quantitative study of Lie algebras with polynomial identities and their universal enveloping algebras. The proof of the main result is also based on the theorem from [\textit{A. A. Kirillov, M. L. Kontsevich} and \textit{A. I. Molev}, Algebras of intermediate growth (Russian), Preprint of the Keldysh Institute of Applied Mathematics, No. 39, Moscow (1983)], stating that a finitely generated associative algebra without zero divisors and of subexponential growth has a classical ring of fractions.
scientific article; zbMATH DE number 1028476