Cluster theory of the coherent Satake category

TL;DR

This paper investigates the coherent Satake category on the affine Grassmannian, revealing its nonsemisimplicity and noncommutative convolution, and establishes its connection to quantum cluster algebras, especially for G=GL_n.

Contribution

It introduces renormalized r-matrices in the coherent Satake category, proves its rigidity, and links these structures to quantum cluster algebras, extending the framework beyond the classical case.

Findings

The coherent Satake category is not semisimple or symmetric.

Construction of canonical renormalized r-matrices satisfying weaker axioms.

The GL_n case forms a monoidal categorification of a quantum cluster algebra.

Abstract

We study the category of G(O)-equivariant perverse coherent sheaves on the affine Grassmannian of 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 loop rotation 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 N=2 field theory may be understood from this point of view.