Work

The crystal groupoid of a compact semisimple Lie group and its flag varieties

Year: 2026

Type: preprint

Abstract: We introduce the crystal groupoid of a compact semisimple Lie group $K$; it is the Deaconu-Renault groupoid of a higher-rank analogue of a subshift of finite type on an inverse limit of Kashiwara crystals. We show that its Steinberg algebra and its $C^*$-algebra realise the crystal limits of the polynomial function algebra of $K$ and its $C^*$-envelope, respectively, in the sense of Matassa and the second author. We show that the orbit space of the crystal groupoid is in bijection with the Weyl group of $K$, with topology determined by the Bruhat order on the Weyl group. We also define a crystal groupoid for flag manifolds and related Poisson homogeneous spaces, and prove the analogous structural results.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.01208

Authors: Sims, Aidan, Yuncken, Robert

Institutions: Centre National de la Recherche Scientifique, Institut Élie Cartan de Lorraine, UNSW Sydney, Université de Lorraine

Language: English

Topic: Advanced Operator Algebra Research

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Stanley Medical Research Institute, Agence Nationale de la Recherche, University of Wollongong

Awards: Operator algebras and representation theory of Lie groups and groupoids, Crossroads in the theory of compact quantum groups, G7080048889

OpenAlex ID: W7215799713

Cites: 19

Cited by:

Related to: 10