Convex topological algebras via linear vector fields and Cuntz algebras

TL;DR

This paper constructs embeddings of certain Lie algebras and topological algebras into linear vector fields and Cuntz algebras, providing new realizations and examples including twisted Heisenberg-Virasoro and Schrödinger-Virasoro algebras.

Contribution

It introduces a method to realize Lie algebras with biorthogonal systems as linear vector fields and embeds topological algebras into Cuntz algebras, expanding algebraic representation theory.

Findings

Explicit embeddings for Lie algebras with biorthogonal systems.

Examples including twisted Heisenberg-Virasoro and Schrödinger-Virasoro algebras.

Embedding of arbitrary locally convex topological algebras into Cuntz algebras.

Abstract

Realization by linear vector fields is constructed for any Lie algebra which admits a biorthogonal system and for its any suitable representation. The embedding into Lie algebras of linear vector fields is analogous to the classical Jordan-Schwinger map. A number of examples of such Lie algebras of linear vector fields is computed. In particular, we obtain examples of the twisted Heisenberg-Virasoro Lie algebra and the Schr\"odinger-Virasoro Lie algebras among others. More generally, we construct an embedding of an arbitrary locally convex topological algebra into the Cuntz algebra.