Analyzing the Weyl Construction for Dynamical Cartan Subalgebras

Anna Duwenig, Elizabeth Gillaspy, Rachael Norton

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


When the reduced twisted C-algebra Cr(G, c) of a non-principal groupoid G admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of Cr(G, c). In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid S of G. In this paper, we study the relationship between the original groupoids S, G and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum B of the Cartan subalgebra Cr(S, c). We then show that the quotient groupoid G/S acts on B, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map G → G/S admits a continuous section, then the Weyl twist is also given by an explicit continuous 2-cocycle on G/S × B.

Original languageEnglish
Pages (from-to)15721-15755
Number of pages35
JournalInternational Mathematics Research Notices
Issue number20
StatePublished - Oct 1 2022


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