Contractions of group representations via geometric quantisation

TL;DR

This paper introduces a geometric quantization framework for contracting unitary duals of Lie groups, providing explicit conditions and constructions for contractions, exemplified by SU(2) to H, applicable to matrix Lie groups.

Contribution

It develops a general geometric quantization approach for group contraction, with explicit cocycle conditions and constructions for matrix Lie groups.

Findings

Explicit contraction of SU(2) into H demonstrated.

Cocycle conditions for contractability established.

Framework applicable to all matrix Lie groups with diagonal contraction matrices.

Abstract

We propose a general framework to contract unitary dual of Lie groups via holomorphic quantization of their co-adjoint orbits. The sufficient condition for the contractability of a representation is expressed via cocycles on coadjoint orbits. This condition is checked explicitly for the contraction of ${\mathrm SU}_2$ into $\mathbb{H}$. The main tool is the geometric quantization. We construct two types of contractions that can be implemented on every matrix Lie group with diagonal contraction matrix.