Trihedral Soergel bimodules

Abstract

The quantum Satake correspondence relates dihedral Soergel bimodules to the semisimple quotient of the quantum sl2 representation category. It also establishes a precise relation between the simple transitive 2-representations of both monoidal categories, which are indexed by bicolored ADE Dynkin diagrams.Using the quantum Satake correspondence between affine A2 Soergel bimodules and the semisimple quotient of the quantum sl3 representation category, we introduce trihedral Hecke algebras and Soergel bimodules, generalizing dihedral Hecke algebras and Soergel bimodules. These have their own Kazhdan-Lusztig combinatorics, simple transitive 2-representations corresponding to tricolored generalized ADE Dynkin diagrams.This section is mostly a collection of known results, formulated in our notation.2.1. Quantum sl 3 . Throughout, m, n, k, l will denote non-negative integers.

Coidempotent Subcoalgebras and Short Exact Sequences of Finitary 2-Representations

“…The question of how to define a 2-categorical analogue of these was answered in [MM5], where the notion of simple transitive 2-representations was defined and a Jordan–Hölder theory for finitary 2-categories was provided. Since then, there has been considerable effort to classify simple transitive 2-representations for certain classes of finitary 2-categories; see for example [MM5, MaMa, Zi, Zi2, Zh, KMMZ, MT, MMMZ, MMZ2, MMMT, MMMTZ].…”

Two-color Soergel Calculus and Simple Transitive 2-representations

“…There should be a similar story for sl >2 , and we hope that our paper will help to develop it. (See [MMMT18] for some first steps in this direction. )…”

Untitled

“…Other versions of 2-representations of Soergel bimodules might also be useful, see e.g. [MT19] or [MMMT20].…”