Categorification and Higher Representation Theory
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 repre...
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| Hauptverfasser: | , |
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| Format: | E-Book Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
2017
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| Ausgabe: | 1 |
| Schriftenreihe: | Contemporary Mathematics |
| Schlagworte: | |
| ISBN: | 9781470424602, 1470424606 |
| ISSN: | 0271-4132, 1098-3627 |
| Online-Zugang: | Volltext |
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Inhaltsangabe:
- Rational Cherednik algebras and categorification -- Categorical actions on unipotent representations of finite classical groups -- Categorical actions and crystals -- On the 2-linearity of the free group -- The Blanchet-Khovanov algebras -- Generic character sheaves on groups over <inline-formula content-type="math/mathml"> k [ ϵ ] / ( ϵ r ) \mathbf k[\epsilon ]/(\epsilon ^r) </inline-formula> -- Integral presentations of quantum lattice Heisenberg algebras -- Categorification at prime roots of unity and hopfological finiteness -- Folding with Soergel bimodules -- The p-canonical basis for Hecke algebras
- 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_{ }
- 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
- 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_{ }
- 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