Defining relations for minimal unitary quantum affine W-algebras

TL;DR

This paper establishes the structure and classification of unitary modules over minimal quantum affine W-algebras, providing explicit relations and a comprehensive list of irreducible modules at non-critical levels.

Contribution

It introduces the defining relations for unitary simple minimal quantum affine W-algebras and classifies their irreducible positive energy modules.

Findings

Any unitary highest weight module descends to the simple quotient.

Explicit defining relations for the unitary simple minimal quantum affine W-algebras.

Complete classification of irreducible positive energy modules.

Abstract

We prove that any unitary highest weight module over a universal minimal quantum affine $W$-algebra at non-critical level descends to its simple quotient. We find the defining relations of the unitary simple minimal quantum affine $W$-algebras and the list of all their irreducible positive energy modules. We also classify all irreducible highest weight modules for the simple affine vertex algebras in the cases when the associated simple minimal $W$-algebra is unitary.