Sofic Lie Algebras

TL;DR

This paper introduces the concept of soficity for Lie algebras, providing equivalent definitions, characterizing subexponential growth Lie algebras as sofic, and establishing connections with their universal enveloping algebras, along with explicit examples.

Contribution

It defines soficity for Lie algebras, proves key characterizations, and constructs explicit almost representations for important examples.

Findings

Lie algebras of subexponential growth are sofic

Soficity over characteristic 0 fields is equivalent to the soficity of their universal enveloping algebras

Explicit almost representations for Witt and Virasoro algebras

Abstract

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