Unitary highest weight modules of locally affine Lie algebras

TL;DR

This paper studies a class of integrable highest weight modules for locally affine Lie algebras, constructing unitary representations and classifying minimal locally affine Lie algebras using root system classifications.

Contribution

It introduces a construction of integrable highest weight modules for locally affine Lie algebras and classifies minimal cases via root system analysis.

Findings

Constructed integrable representations for each non-vanishing integral weight.

Established unitary representations over complex numbers.

Classified minimal locally affine Lie algebras using root system data.

Abstract

Locally affine Lie algebras are generalizations of affine Kac--Moody algebras with Cartan subalgebras of infinite rank whose root system is locally affine. In this note we study a class of representations of locally affine algebras generalizing integrable highest weight modules. In particular, we construct such an integrable representation for each integral weight not vanishing on the center and show that, over the complex numbers, we thus obtain unitary representations w.r.t. a unitary real form. We also use Yoshii's recent classification of locally affine root systems to derive a classification of so-called minimal locally affine Lie algebras and give realizations as twisted loop algebras.