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Categorification and Higher Representation Theory.

By: Contributor(s): Material type: TextTextSeries: Contemporary Mathematics SeriesPublisher: Providence : American Mathematical Society, 2017Copyright date: ©2017Edition: 1st edDescription: 1 online resource (376 pages)Content type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9781470436896
Subject(s): Genre/Form: Additional physical formats: Print version:: Categorification and Higher Representation TheoryDDC classification:
  • 512/.62
LOC classification:
  • QA169.C384 2017
Online resources:
Contents:
Cover -- Title page -- Contents -- Preface -- Rational Cherednik algebras and categorification -- 1. Introduction -- 2. Rational Cherednik algebras and categories -- 3. Cyclotomic categories and categorification -- 4. Supports of simple modules -- 5. Category equivalences and multiplicities -- References -- Categorical actions on unipotent representations of finite classical groups -- Introduction -- 1. Categorical representations -- 1.1. Rings and categories -- 1.2. Kac-Moody algebras of type and their representations -- 1.2.1. Lie algebra associated with a quiver -- 1.2.2. Integrable representations -- 1.2.3. Quantized enveloping algebras -- 1.3. Categorical representations on abelian categories -- 1.3.1. Affine Hecke algebras and representation data -- 1.3.2. Categorical representations -- 1.4. Minimal categorical representations -- 1.5. Crystals -- 1.6. Perfect bases -- 1.7. Derived equivalences -- 2. Representations on Fock spaces -- 2.1. Combinatorics of -partitions -- 2.1.1. Partitions and -partitions -- 2.1.2. Residues and content -- 2.1.3. -cores and -quotients -- 2.2. Fock spaces -- 2.3. Charged Fock spaces -- 2.3.1. The \frakg-action on the Fock space -- 2.3.2. The crystal of the Fock space -- 3. Unipotent representations -- 3.1. Basics -- 3.2. Unipotent -modules -- 3.3. Unipotent \myk -modules and ℓ-blocks -- 3.4. Harish-Chandra series -- 4. Finite unitary groups -- 4.1. Definition -- 4.2. The representation datum on -mod -- 4.3. The categories of unipotent modules \scrU_{ } and \scrU_{\myk} -- 4.3.1. The category \scrU_{ } -- 4.3.2. The category \scrU_{\myk} -- 4.3.3. Blocks of \scrU_{\myk} -- 4.3.4. The weak Harish-Chandra series -- 4.4. The \frakg_{∞}-representation on \scrU_{ } -- 4.4.1. Action of and -- 4.4.2. The ramified Hecke algebra -- 4.4.3. Parametrization of the weak Harish-Chandra series of \scrU_{ }.
4.4.4. The \frakg_{∞}-representation on \scrU_{ } -- 4.5. The \frakgₑ-representation on \scrU_{\myk}. -- 4.5.1. The Lie algebras \frakgₑ and \frakg_{ ,∘} -- 4.5.2. The \frakg'ₑ-action on [\scrU_{\myk}] -- 4.5.3. The \frakgₑ-action on [\scrU_{\myk}] -- 4.5.4. The \frakgₑ-action on \scrU_{\myk} -- 4.6. Derived equivalences of blocks of \scrU_{\myk} -- 4.6.1. Characterization of the blocks of \scrU_{\myk} -- 4.6.2. Derived equivalences of blocks of \scrU_{\myk} -- 4.7. The crystals of \scrU_{ } and \scrU_{\myk} -- 4.7.1. Crystals and Harish-Chandra series -- 4.7.2. Comparison of the crystals -- 5. The representation of the Heisenberg algebra on \scrU_{\myk} -- 5.1. The Heisenberg action on [\scrU_{\myk}] -- 5.1.1. The Heisenberg algebra -- 5.1.2. The Heisenberg action on \bfF( ) -- 5.1.3. The Heisenberg action on [\scrU_{\myk}] -- 5.2. The modular Harish-Chandra series of _{ } -- 5.2.1. The unipotent modules of _{ } -- 5.2.2. The modular Steinberg character and Harish-Chandra series -- 5.3. The Heisenberg functors -- 5.4. The categorification of the Heisenberg action on [\scrU_{\myk}] -- 5.5. Cuspidal modules and highest weight vectors -- 5.5.1. The parameters of the ramified Hecke algebras -- 5.5.2. The classification of cuspidal unipotent modules -- 5.5.3. Cuspidal modules and FLOTW -partitions -- 6. Types B and C -- 6.1. Definitions -- 6.1.1. Odd-dimensional orthogonal groups -- 6.1.2. Symplectic groups -- 6.2. The representation datum on -mod -- 6.3. The categories of unipotent modules \scrU_{ } and \scrU_{\myk} -- 6.3.1. Parametrization by symbols -- 6.3.2. The unipotent modules over -- 6.3.3. The unipotent modules over \myk -- 6.3.4. The unipotent blocks -- 6.4. The \frakg_{∞}-representation on \scrU_{ } -- 6.4.1. The ramified Hecke algebra -- 6.4.2. The \frakg_{∞}-representation on \scrU_{ }.
6.5. The \frakg_{2 }-representation on \scrU_{\myk} -- 6.5.1. The \frakg'_{2 }-representation on \scrU_{\myk} -- 6.5.2. The \frakg_{2 }-representation on \scrU_{\myk} in the linear prime case -- 6.5.3. Combinatorics of -cohooks and -cocores -- 6.5.4. The weight of a symbol -- 6.5.5. The \frakg_{2 }-representation on \scrU_{\myk} in the unitary case -- 6.5.6. Determination of the ramified Hecke algebras -- 6.6. Derived equivalences -- 6.7. The crystal of \scrU_{\myk} -- 6.7.1. Ordering symbols -- 6.7.2. Parametrization of unipotent modules -- 6.7.3. Comparison of the crystals -- References -- Categorical actions and crystals -- 1. Introduction -- 2. Locally Schurian categories -- 3. Kac-Moody 2-categories -- 4. Categorical actions and crystals -- References -- On the 2-linearity of the free group -- 1. Introduction -- 2. The Free group -- 3. Ping pong and dual ping pong -- 4. A 2-representation the free group -- 5. 2-linearity via ping pong and the ̃ -grading on \A -- 6. The Bessis monoid, the ⃗ -grading on \A, and dual ping pong -- 7. Metrics on _{ } from homological algebra -- References -- The Blanchet-Khovanov algebras -- 1. Introduction -- 2. gl2-foams and gl2-web algebras -- 2.1. Webs, foams and TQFTs -- 2.2. Blanchet's singular TQFT construction -- 2.3. An action of the quantum group Udot(glinfty) -- 2.4. gl2-web algebras -- 2.5. Web bimodules -- 3. Blanchet-Khovanov algebras -- 3.1. Combinatorics of arc diagrams -- 3.2. The Blanchet-Khovanov algebras as graded K-vector spaces -- 3.3. Multiplication of the Blanchet-Khovanov algebra -- 3.4. Bimodules for Blanchet-Khovanov algebras -- 4. Equivalences -- 4.1. Some useful lemmas -- 4.2. An action of the quantum group Udot(glinfty) and arc diagrams -- 4.3. The cup basis -- 4.4. Proof of the main result -- 4.5. The proof of the graded isomorphism -- Acknowledgements -- References.
Generic character sheaves on groups over [ ]/( ^{ }) -- Introduction -- 1. The complex -- 2. The cases =2 and =4 -- 3. The case =3 -- 4. A comparison of two complexes -- Acknowledgement -- References -- Integral presentations of quantum lattice Heisenberg algebras -- 1. Introduction -- 2. Hopf algebras, Hopf pairings, and the Heisenberg double -- 3. The ring of symmetric functions -- 4. Symmetric and exterior algebras -- 5. Presentations of quantum lattice Heisenberg algebras -- Acknowledgements -- References -- Categorification at prime roots of unity and hopfological finiteness -- 1. Introduction -- 2. The small quantum group -- 3. A categorical braid group action at a prime root of unity -- 4. A categorification of quantum ₂ at prime roots of unity -- 5. A categorification of the Jones-Wenzl projector -- References -- Folding with Soergel bimodules -- 1. Introduction -- 2. Hecke algebras and unequal parameters -- 3. Equivariant categories and weighted Grothendieck groups -- 4. Soergel bimodules -- 5. Folding -- Acknowledgments -- References -- The p-canonical basis for Hecke algebras -- 1. Introduction -- 1.1. -- 1.2. -- 1.3. -- 1.4. Structure of the Paper: -- Acknowledgements -- 2. Background -- 2.1. Coxeter Systems and Based Root Data -- 2.2. The Hecke Algebra -- 2.3. Soergel Calculus -- 2.4. Light Leaves and Double Leaves -- 2.5. The Diagrammatic Category: Properties -- 3. The p-Canonical Basis and Intersection Forms -- 3.1. Calculations in the nil Hecke Ring -- 4. First Properties of the p-Canonical Basis -- 4.1. The Geometric Satake and the p-canonical Basis -- 5. Examples -- 5.1. Type B2 -- 5.2. Type G2 -- 5.3. Type A1 -- 5.4. Types B3 and C3 -- 5.5. Type D4 -- 5.6. Type An -- References -- Back Cover.
Summary: The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorified representation theory, or higher representation theory, aims to understand a new level of structure present in representation theory. Rather than studying actions of algebras on vector spaces where algebra elements act by linear endomorphisms of the vector space, higher representation theory describes the structure present when algebras act on categories, with algebra elements acting by functors. The new level of structure in higher representation theory arises by studying the natural transformations between functors. This enhanced perspective brings into play a powerful new set of tools that deepens our understanding of traditional representation theory.This volume exhibits some of the current trends in higher representation theory and the diverse techniques that are being employed in this field with the aim of showcasing the many applications of higher representation theory.The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.
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Cover -- Title page -- Contents -- Preface -- Rational Cherednik algebras and categorification -- 1. Introduction -- 2. Rational Cherednik algebras and categories -- 3. Cyclotomic categories and categorification -- 4. Supports of simple modules -- 5. Category equivalences and multiplicities -- References -- Categorical actions on unipotent representations of finite classical groups -- Introduction -- 1. Categorical representations -- 1.1. Rings and categories -- 1.2. Kac-Moody algebras of type and their representations -- 1.2.1. Lie algebra associated with a quiver -- 1.2.2. Integrable representations -- 1.2.3. Quantized enveloping algebras -- 1.3. Categorical representations on abelian categories -- 1.3.1. Affine Hecke algebras and representation data -- 1.3.2. Categorical representations -- 1.4. Minimal categorical representations -- 1.5. Crystals -- 1.6. Perfect bases -- 1.7. Derived equivalences -- 2. Representations on Fock spaces -- 2.1. Combinatorics of -partitions -- 2.1.1. Partitions and -partitions -- 2.1.2. Residues and content -- 2.1.3. -cores and -quotients -- 2.2. Fock spaces -- 2.3. Charged Fock spaces -- 2.3.1. The \frakg-action on the Fock space -- 2.3.2. The crystal of the Fock space -- 3. Unipotent representations -- 3.1. Basics -- 3.2. Unipotent -modules -- 3.3. Unipotent \myk -modules and ℓ-blocks -- 3.4. Harish-Chandra series -- 4. Finite unitary groups -- 4.1. Definition -- 4.2. The representation datum on -mod -- 4.3. The categories of unipotent modules \scrU_{ } and \scrU_{\myk} -- 4.3.1. The category \scrU_{ } -- 4.3.2. The category \scrU_{\myk} -- 4.3.3. Blocks of \scrU_{\myk} -- 4.3.4. The weak Harish-Chandra series -- 4.4. The \frakg_{∞}-representation on \scrU_{ } -- 4.4.1. Action of and -- 4.4.2. The ramified Hecke algebra -- 4.4.3. Parametrization of the weak Harish-Chandra series of \scrU_{ }.

4.4.4. The \frakg_{∞}-representation on \scrU_{ } -- 4.5. The \frakgₑ-representation on \scrU_{\myk}. -- 4.5.1. The Lie algebras \frakgₑ and \frakg_{ ,∘} -- 4.5.2. The \frakg'ₑ-action on [\scrU_{\myk}] -- 4.5.3. The \frakgₑ-action on [\scrU_{\myk}] -- 4.5.4. The \frakgₑ-action on \scrU_{\myk} -- 4.6. Derived equivalences of blocks of \scrU_{\myk} -- 4.6.1. Characterization of the blocks of \scrU_{\myk} -- 4.6.2. Derived equivalences of blocks of \scrU_{\myk} -- 4.7. The crystals of \scrU_{ } and \scrU_{\myk} -- 4.7.1. Crystals and Harish-Chandra series -- 4.7.2. Comparison of the crystals -- 5. The representation of the Heisenberg algebra on \scrU_{\myk} -- 5.1. The Heisenberg action on [\scrU_{\myk}] -- 5.1.1. The Heisenberg algebra -- 5.1.2. The Heisenberg action on \bfF( ) -- 5.1.3. The Heisenberg action on [\scrU_{\myk}] -- 5.2. The modular Harish-Chandra series of _{ } -- 5.2.1. The unipotent modules of _{ } -- 5.2.2. The modular Steinberg character and Harish-Chandra series -- 5.3. The Heisenberg functors -- 5.4. The categorification of the Heisenberg action on [\scrU_{\myk}] -- 5.5. Cuspidal modules and highest weight vectors -- 5.5.1. The parameters of the ramified Hecke algebras -- 5.5.2. The classification of cuspidal unipotent modules -- 5.5.3. Cuspidal modules and FLOTW -partitions -- 6. Types B and C -- 6.1. Definitions -- 6.1.1. Odd-dimensional orthogonal groups -- 6.1.2. Symplectic groups -- 6.2. The representation datum on -mod -- 6.3. The categories of unipotent modules \scrU_{ } and \scrU_{\myk} -- 6.3.1. Parametrization by symbols -- 6.3.2. The unipotent modules over -- 6.3.3. The unipotent modules over \myk -- 6.3.4. The unipotent blocks -- 6.4. The \frakg_{∞}-representation on \scrU_{ } -- 6.4.1. The ramified Hecke algebra -- 6.4.2. The \frakg_{∞}-representation on \scrU_{ }.

6.5. The \frakg_{2 }-representation on \scrU_{\myk} -- 6.5.1. The \frakg'_{2 }-representation on \scrU_{\myk} -- 6.5.2. The \frakg_{2 }-representation on \scrU_{\myk} in the linear prime case -- 6.5.3. Combinatorics of -cohooks and -cocores -- 6.5.4. The weight of a symbol -- 6.5.5. The \frakg_{2 }-representation on \scrU_{\myk} in the unitary case -- 6.5.6. Determination of the ramified Hecke algebras -- 6.6. Derived equivalences -- 6.7. The crystal of \scrU_{\myk} -- 6.7.1. Ordering symbols -- 6.7.2. Parametrization of unipotent modules -- 6.7.3. Comparison of the crystals -- References -- Categorical actions and crystals -- 1. Introduction -- 2. Locally Schurian categories -- 3. Kac-Moody 2-categories -- 4. Categorical actions and crystals -- References -- On the 2-linearity of the free group -- 1. Introduction -- 2. The Free group -- 3. Ping pong and dual ping pong -- 4. A 2-representation the free group -- 5. 2-linearity via ping pong and the ̃ -grading on \A -- 6. The Bessis monoid, the ⃗ -grading on \A, and dual ping pong -- 7. Metrics on _{ } from homological algebra -- References -- The Blanchet-Khovanov algebras -- 1. Introduction -- 2. gl2-foams and gl2-web algebras -- 2.1. Webs, foams and TQFTs -- 2.2. Blanchet's singular TQFT construction -- 2.3. An action of the quantum group Udot(glinfty) -- 2.4. gl2-web algebras -- 2.5. Web bimodules -- 3. Blanchet-Khovanov algebras -- 3.1. Combinatorics of arc diagrams -- 3.2. The Blanchet-Khovanov algebras as graded K-vector spaces -- 3.3. Multiplication of the Blanchet-Khovanov algebra -- 3.4. Bimodules for Blanchet-Khovanov algebras -- 4. Equivalences -- 4.1. Some useful lemmas -- 4.2. An action of the quantum group Udot(glinfty) and arc diagrams -- 4.3. The cup basis -- 4.4. Proof of the main result -- 4.5. The proof of the graded isomorphism -- Acknowledgements -- References.

Generic character sheaves on groups over [ ]/( ^{ }) -- Introduction -- 1. The complex -- 2. The cases =2 and =4 -- 3. The case =3 -- 4. A comparison of two complexes -- Acknowledgement -- References -- Integral presentations of quantum lattice Heisenberg algebras -- 1. Introduction -- 2. Hopf algebras, Hopf pairings, and the Heisenberg double -- 3. The ring of symmetric functions -- 4. Symmetric and exterior algebras -- 5. Presentations of quantum lattice Heisenberg algebras -- Acknowledgements -- References -- Categorification at prime roots of unity and hopfological finiteness -- 1. Introduction -- 2. The small quantum group -- 3. A categorical braid group action at a prime root of unity -- 4. A categorification of quantum ₂ at prime roots of unity -- 5. A categorification of the Jones-Wenzl projector -- References -- Folding with Soergel bimodules -- 1. Introduction -- 2. Hecke algebras and unequal parameters -- 3. Equivariant categories and weighted Grothendieck groups -- 4. Soergel bimodules -- 5. Folding -- Acknowledgments -- References -- The p-canonical basis for Hecke algebras -- 1. Introduction -- 1.1. -- 1.2. -- 1.3. -- 1.4. Structure of the Paper: -- Acknowledgements -- 2. Background -- 2.1. Coxeter Systems and Based Root Data -- 2.2. The Hecke Algebra -- 2.3. Soergel Calculus -- 2.4. Light Leaves and Double Leaves -- 2.5. The Diagrammatic Category: Properties -- 3. The p-Canonical Basis and Intersection Forms -- 3.1. Calculations in the nil Hecke Ring -- 4. First Properties of the p-Canonical Basis -- 4.1. The Geometric Satake and the p-canonical Basis -- 5. Examples -- 5.1. Type B2 -- 5.2. Type G2 -- 5.3. Type A1 -- 5.4. Types B3 and C3 -- 5.5. Type D4 -- 5.6. Type An -- References -- Back Cover.

The emergent mathematical philosophy of categorification is reshaping our view of modern mathematics by uncovering a hidden layer of structure in mathematics, revealing richer and more robust structures capable of describing more complex phenomena. Categorified representation theory, or higher representation theory, aims to understand a new level of structure present in representation theory. Rather than studying actions of algebras on vector spaces where algebra elements act by linear endomorphisms of the vector space, higher representation theory describes the structure present when algebras act on categories, with algebra elements acting by functors. The new level of structure in higher representation theory arises by studying the natural transformations between functors. This enhanced perspective brings into play a powerful new set of tools that deepens our understanding of traditional representation theory.This volume exhibits some of the current trends in higher representation theory and the diverse techniques that are being employed in this field with the aim of showcasing the many applications of higher representation theory.The companion volume (Contemporary Mathematics, Volume 684) is devoted to categorification in geometry, topology, and physics.

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