Trihedral Soergel bimodules

Marco MacKaay; Volodymyr Mazorchuk; Vanessa Miemietz; Daniel Tubbenhauer

doi:10.4064/fm566-3-2019

Trihedral Soergel bimodules

Marco MacKaay, Volodymyr Mazorchuk, Vanessa Miemietz, Daniel Tubbenhauer

Algebra and Combinatorics

Research output: Contribution to journal › Article › peer-review

3 Citations (Scopus)

34 Downloads (Pure)

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 A ₂ 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.

Original language	English
Pages (from-to)	219-300
Number of pages	82
Journal	Fundamenta Mathematicae
Volume	248
Issue number	3
Early online date	19 Sep 2019
DOIs	https://doi.org/10.4064/fm566-3-2019
Publication status	Published - Jan 2020

Keywords

2-representation theory
Hecke algebras
Quantum groups and their fusion categories
Soergel bimodules
Zigzag algebras

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10.4064/fm566-3-2019Licence: Other

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https://arxiv.org/abs/1804.08920

Cite this

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title = "Trihedral Soergel bimodules",

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AU - MacKaay, Marco

AU - Mazorchuk, Volodymyr

AU - Miemietz, Vanessa

AU - Tubbenhauer, Daniel

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N2 - 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 A 2 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.

AB - 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 A 2 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.

KW - 2-representation theory

KW - Hecke algebras

KW - Quantum groups and their fusion categories

KW - Soergel bimodules

KW - Zigzag algebras

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DO - 10.4064/fm566-3-2019

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EP - 300

JO - Fundamenta Mathematicae

JF - Fundamenta Mathematicae

SN - 0016-2736

IS - 3

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Trihedral Soergel bimodules

Abstract

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Vanessa Miemietz

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