Linearly sofic Lie algebras

Abstract

We introduce and study linear soficity for Lie algebras, modeled after linear soficity in associative algebras. We introduce equivalent definitions of linear soficity, one involving metric ultraproducts and the other involving almost representations. We prove that any Lie algebra of subexponential growth is linearly sofic. We also prove that a Lie algebra over a field of characteristic 0 is linearly sofic if and only if its universal enveloping algebra is linearly sofic. As examples, we give explicit families of almost representations for the Witt and Virasoro algebras. Finally, we also show that the restricted universal enveloping algebra of a restricted linearly sofic Lie algebra is also linearly sofic.