Multiplet classification for SU(n,n)

TL;DR

This paper systematically constructs and classifies invariant differential operators and minimal representations for the non-compact algebras su(n,n), providing explicit multiplet structures for n=2,3,4, with broader applicability to related algebras.

Contribution

It offers the first explicit classification of multiplets and minimal representations for su(n,n) for n=2,3,4, and extends results to related algebra classes via parabolic relations.

Findings

Explicit multiplet structures for su(2,2), su(3,3), su(4,4)

Construction of minimal representations for these algebras

Extension of classification results to sl(2n,R) and su*(4k)

Abstract

In the present paper we review our project of systematic construction of invariant differential operators on the example of the non-compact algebras su(n,n) for n=2,3,4. We give explicitly the main multiplets of indecomposable elementary representations and some reduced multiplets. We give explicitly the minimal representations. Due to the recently established parabolic relations the multiplet classification results are valid also for the algebras sl(2n,R) and when n=2k for the algebras su*(4k) with suitably chosen maximal parabolic subalgebras.