PreprintCC BY 4.0arXiv
The crystal groupoid of a compact semisimple Lie group and its flag varieties
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.
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BibTeX
@misc{sims2026crystal,
title = {{The crystal groupoid of a compact semisimple Lie group and its flag varieties}},
author = {Aidan Sims and Robert Yuncken},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01208},
eprint = {2610.01208},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01208},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The crystal groupoid of a compact semisimple Lie group and its flag varieties
AU - Aidan Sims
AU - Robert Yuncken
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01208
UR - https://arxiv.org/abs/2610.01208
AB - 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.
ER -