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Cluster theory of the coherent Satake category


Authors: Sabin Cautis and Harold Williams
Journal: J. Amer. Math. Soc. 32 (2019), 709-778
MSC (2010): Primary 13F60, 22E67; Secondary 14F05
DOI: https://doi.org/10.1090/jams/918
Published electronically: April 10, 2019
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Abstract: We study the category of $ G(\mathcal {O})$-equivariant perverse coherent
sheaves on the affine Grassmannian $ \mathrm {Gr}_G$. This coherent Satake category is not semisimple and its convolution product is not symmetric, in contrast with the usual constructible Satake category. Instead, we use the Beilinson-Drinfeld Grassmannian to construct renormalized $ r$-matrices. These are canonical nonzero maps between convolution products which satisfy axioms weaker than those of a braiding.

We also show that the coherent Satake category is rigid, and that together these results strongly constrain its convolution structure. In particular, they can be used to deduce the existence of (categorified) cluster structures. We study the case $ G = GL_n$ in detail and prove that the $ \mathbb{G}_m$-equivariant coherent Satake category of $ GL_n$ is a monoidal categorification of an explicit quantum cluster algebra.

More generally, we construct renormalized $ r$-matrices in any monoidal category whose product is compatible with an auxiliary chiral category, and explain how the appearance of cluster algebras in 4d $ \mathcal {N}=2$ field theory may be understood from this point of view.


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Additional Information

Sabin Cautis
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, British Columbia V6T 1Z2, Canada
Email: cautis@math.ubc.ca

Harold Williams
Affiliation: Department of Mathematics, University of California, Davis, California 95616
Email: hwilliams@math.ucdavis.edu

DOI: https://doi.org/10.1090/jams/918
Received by editor(s): February 17, 2018
Received by editor(s) in revised form: January 11, 2019
Published electronically: April 10, 2019
Additional Notes: The first author was supported by an NSERC Discovery/accelerator grant.
The second author was supported by NSF Postdoctoral Fellowship DMS-1502845 and NSF grant DMS-1702489.
Article copyright: © Copyright 2019 American Mathematical Society