Representation Theory, Mathematical Physics, and Integrable Systems: In Honor of Nicolai Reshetikhin

Preface
1 Quantum Groups
2 Quantum Integrable Systems
3 Topology
4 Representation Theory and Combinatorics
5 Poisson Algebras, BV Theory, and Quantization
6 Other Works
References
Contents
Examples of Finite-Dimensional Pointed Hopf Algebras in Positive Characteristic
1 Introduction
1.1 Overview
1.2 The Main Result
1.3 Contents of the Paper
2 Preliminaries
2.1 Conventions
2.2 Yetter-Drinfeld Modules
2.3 Nichols Algebras
3 Blocks
3.1 The Jordan Plane
3.2 The Super Jordan Plane
3.3 Realizations
3.4 Exhaustion in Rank 2
4 One Block and One Point
4.1 The Setting and the Statement
4.2 Weak Interaction
4.3 The Presentation by Generators and Relations
4.3.1 The Nichols Algebra B (L( 1, G ))
4.3.2 The Nichols Algebra B (L( -1, G ))
4.3.3 The Nichols Algebra B (L-( 1, G ))
4.3.4 The Nichols Algebra B (L-( -1, G ))
4.3.5 The Nichols Algebra B (L( ω, 1))
4.4 Mild Interaction
4.5 Realizations
5 One Block and Several Points
5.1 The Setting and the Main Result
5.2 The Presentation of the Nichols Algebras
5.2.1 The Nichols Algebra B (L(A(1"026A30C 0)1; r)), rGN, N≥3
5.2.2 The Nichols Algebra B (L(A(1"026A30C 0)2; ω))
5.2.3 The Nichols Algebra B (L(A(1"026A30C 0)3; ω))
5.2.4 The Nichols Algebra B (L(A(2"026A30C 0)1; ω))
5.2.5 The Nichols Algebra B (L(D(2"026A30C 1); ω))
5.2.6 The Nichols Algebra B (L(A2, 2))
5.2.7 The Nichols Algebra B (L(Aθ- 1))
5.3 Realizations
6 Several Blocks, One Point
6.1 Realizations
7 A Pale Block and a Point
7.1 The Block Has ε= 1
7.2 The Block Has ε= -1
7.2.1 Case 1: q"0365q12 = 1
7.2.2 Case 2: q"0365q12=-1
7.3 Realizations
References
Poisson Vertex Algebra Cohomology and Differential Harrison Cohomology
1 Introduction
2 Differential Harrison Cohomology Complex
2.1 Hochschild Cohomology Complex
2.2 Monotone Permutations
2.3 Differential Harrison Cohomology Complex
3 The Classical Operad and PVA Cohomology
3.1 Symmetric Group Actions
3.2 Composition of Permutations and Shuffles
3.3 n-Graphs
3.4 Lie Conformal Algebras and Poisson Vertex Algebras
3.5 Operads
3.6 The Z-graded Lie Superalgebra Associated with an Operad
3.7 The Classical Operad Pcl BDSHK19
3.8 PVA Cohomology BDSHK19
4 Relation Between PVA and Differential Harrison Cohomology Complexes
4.1 Main Theorem
4.2 Lines
4.3 Connected Lines
4.4 Relation Between the Symmetry Property and Harrison's Conditions
4.5 Relation Between `3́9`42`"̇613A``45`47`"603AadX and the Hochschild Differential
4.6 Proof of Theorem 4.1
References
Theta Invariants of Lens Spaces via the BV-BFV Formalism
1 Introduction
2 Perturbative Quantization of Chern-Simons Theory
2.1 Chern-Simons Theory
2.2 Perturbative Quantization of Gauge Theories
2.3 Perturbative Chern-Simons Theory and Invariants of 3-Manifolds and Links
3 Review of BV-BFV Quantization
3.1 BV Formalism
3.1.1 Effective Action and Residual Fields
3.2 BV-BFV Extension on Manifolds with Boundary
3.2.1 Classical Case
3.2.2 Quantization
4 Split Chern-Simons Theory on the Solid Torus
4.1 Split Chern-Simons Theory
4.2 Polarization
4.3 Residual Fields and Fluctuations
4.4 Axial Gauge Propagator
4.5 The State
4.5.1 Feynman Graphs and Rules
4.5.2 Regularization
5 Effective Action on the Solid Torus
5.1 Zero-Point Contribution
5.2 One-Point Contribution
5.2.1 Possible Diagrams
5.2.2 baa Term
5.2.3 baα Term
5.2.4 bαα Term
5.2.5 aαα Term
5.2.6 ααα Term
5.3 2-Point Tree Contribution
5.4 Loop Diagrams
5.4.1 Regularization
5.4.2 Evaluation
6 Gluing of Lens Spaces
6.1 Lens Spaces
6.2 Gluing Perturbative Expansions in BV-BFV
6.3 The Effective Action on M2
6.4 Reducing the Residual Fields
6.4.1 Case p≠0
6.4.2 The Case p=0
7 The Effective Action on Lens Spaces
7.1 Case of a Manin Triple
7.1.1 Pairing Against Order 0 Diagram
7.1.2 Pairing the 1-Point Functions
7.1.3 Reducing the Residual Fields
7.2 The General Case
7.3 Weights of Oriented Theta Graphs on Lens Spaces
7.3.1 Low-Order Diagrams on Rational Homology Spheres
7.3.2 Lens Spaces
8 Comparing to Existing Results
8.1 Kuperberg-Thurston-Lescop Theta Invariants
8.2 Comparison with Weights of Oriented Theta Graphs
9 Conclusions and Outlook
References
Generalized Demazure Modules and Prime Representations in Type Dn
1 Introduction
2 Generalized Demazure Modules
2.1 The Simple Lie Algebra of Type Dn
2.2 The Affine Lie Algebra Dn1
2.3 Highest Weight Modules and Demazure Modules
2.4 Generalized Demazure Modules
2.5 A Presentation of Stable Demazure Modules
2.6 Stable Generalized Demazure Modules
2.7 The Modules V(λ,μ)
2.8 Interlacing Weights
2.9 The Main Theorem
2.10 A First Reduction
2.11 The Root βλ
2.12 The Second Reduction
2.13 Proof of Theorem 2.2
2.14 Proof of Theorem 2.2, the Final Step
2.15 A Character Formula
3 Proof of Propositions 2.1 and 2.2
3.1 Elementary Properties of Interlacing Pairs
3.2 The Set R(λ1,λ2)
3.3 Proof of Proposition 2.1
3.4 The Kernel of ψ
3.5 Proof of Proposition 2.2
3.6 Proof of Proposition 2.2, the Second Step
3.7 Proof of Proposition 2.2, the Final Step
4 Connection with the Representation Theory of Quantum Affine Algebras
4.1 A Multiplicative Monoid of Weights
4.2 Prime Representations
4.3 Graded Limits
4.4 The Connection with the Category Cξ
References
Cylindric Rhombic Tableaux and the Two-Species ASEP on a Ring
1 Introduction
2 The ASEP on a Ring
3 Probabilities for the Two-Species ASEP Using Cylindric Rhombic Tableaux
4 Formulas for Macdonald Polynomials Using Cylindric Rhombic Tableaux
5 The Matrix Ansatz and the Results of Cantini-deGier-Wheeler
6 The Proofs of Theorems 3.17 and 4.9
6.1 Relations Among the Matrices from Definition 5.1
6.2 The Recurrence for Matrix Products
6.3 The Recurrence for Weight Generating Functions of Tableaux
6.4 Symmetries of the Tableaux
7 A Bijection from Cylindric Rhombic Tableaux to Two-Line Queues
References
Macdonald Operators and Quantum Q-Systems for Classical Types
1 Introduction
2 Q-Systems and Quantum Q-Systems
2.1 Weights and Roots
2.2 The Classical Q-Systems for Untwisted Affine ABCD
2.3 Quantum Q-Systems
3 The AN-1 Case Solution: Generalized Macdonald Difference Operators and Quantum Determinants
3.1 Renormalized Quantum Q-System
3.2 The Quantum Determinant
3.3 The Functional Representation of the Quantum Q-System
3.4 The Spherical Double Affine Hecke Algebra
3.5 Raising and Lowering Operators
3.6 Graded Characters in Terms of Difference Operators
4 The Quantum Q-System Conjectures for Types BCD
4.1 Macdonald and van Diejen Operators
4.2 The q-Whittaker Limit
4.3 Discrete Time Evolution by Adjoint Action of the Gaussian
4.4 Other Macdonald Operators via Quantum Determinants
4.5 The Quantum Q-System Conjectures
5 The Raising/Lowering Operator Conjectures for Types BCD
6 The Graded Character Conjectures for Types BCD
7 Conclusion
7.1 Summary: Macdonald Operators and Quantum Cluster Algebra
7.2 Towards Proving the Conjectures
7.3 sDAHA, EHA and the t-Deformation of Quantum Q-Systems
Appendix A: Examples
A.1 Weyl-Invariant Schur Functions
A.2 The B2 Case
A.2.1 M Operators
A.2.2 Dual q-Whittaker Functions
A.3 The C2 Case
A.3.1 The B2C2 Symmetry
A.3.2 M Operators
A.3.3 Dual q-Whittaker Functions
A.4 The sl4 and D3 Cases
A.4.1 The sl4D3 Symmetry
A.4.2 M Operators
A.4.3 Dual q-Whittaker Functions
A.5 The D4 Case
A.5.1 Symmetries
A.5.2 M Operators
A.5.3 Dual q-Whittaker Functions
References
The Meromorphic R-Matrix of the Yangian
1 Introduction
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12 Outline of the Paper
2 The Yangian Y.12em.1emdotteddotteddotted.76dotted.6h(g)
2.1
2.2 The Yangian Y.12em.1emdotteddotteddotted.76dotted.6h(g) drinfeld-yangian-qaffine
2.3
2.4 Alternative Generators of Y0.12em.1emdotteddotteddotted.76dotted.6h(g)
2.5 Shift Automorphism
2.6 PBW Theorem
2.7 The Embedding U(g)Y.12em.1emdotteddotteddotted.76dotted.6h(g)
2.8 Formal Series Filtration
2.9 Rationality
3 The Standard and Drinfeld Coproducts
3.1 Standard Coproduct
3.2 Deformed Drinfeld Tensor Product
3.3 Laurent Expansion of the Deformed Drinfeld Tensor Product
3.4 Deformed Drinfeld Coproduct
4 The Element R-(s)
4.1
4.2 Existence and Uniqueness of R-(s)
4.3 Translation Invariance
4.4 Semiclassical Limit
4.5 Rationality
4.6 Cocycle Equation
4.7 Rank 1 Reduction
4.8 Rank 1 Intertwining Relations
5 The Element R-(s) for sl2
5.1
5.2
5.3 Formula for ω(s,z)
5.4 Formula for ωV1,V2(s,z)
5.5 Formula for R-(s)
6 The Universal and the Meromorphic Abelian R-Matrices of Y.12em.1emdotteddotteddotted.76dotted.6h(g)
6.1 The Endomorphism AV1,V2(s) sachin-valerio-III
6.2 The Meromorphic Abelian R-Matrix of Y.12em.1emdotteddotteddotted.76dotted.6h(g) sachin-valerio-III
6.3 Existence of a Rational Intertwiner
6.4 Non-existence of Rational Commutativity Constraints
6.5 Taylor Expansion of AV1,V2(s)
6.6 Asymptotic Expansion of R0,"3222378 /"3223379 V1,V2(s)
6.7 Properties of R0(s)
6.8 Direct Proof of Theorem 6.7
7 The Universal and the Meromorphic R-Matrices of Y.12em.1emdotteddotteddotted.76dotted.6h(g)
7.1 The Meromorphic R-Matrix
7.2 Existence of a Rational Intertwiner
7.3 Non Existence of Rational Commutativity Constraints
7.4 The Universal R-Matrix
8 Meromorphic Tensor Structures
8.1 Drinfeld Tensor Product
8.2 Standard Tensor Product
8.3 Meromorphic Tensor Structures
9 Relation to the Quantum Loop Algebra Uq(Lg)
9.1 The Functor Γ sachin-valerio-2
9.2 Meromorphic Tensor Structure on Γ sachin-valerio-III
9.3 Tensor Structure with Respect to the Standard Coproducts
9.4 Non Regularity of J"3222378 /"3223379 V1,V2(s)
9.5 The Meromorphic Abelian R-Matrix of Uq(Lg)
9.6 Abelian qDrinfeld–Kohno Theorem
9.7 Meromorphic Braided Tensor Equivalence for Standard Coproducts
A Separation of Points
A.1
A.2 Reduction Step
A.3
A.4
A.5 Proof That J Vanishes
B Uniqueness of the Universal R-Matrix
B.1
B.2
References
On Spectral Cover Equations in Simpson Integrable Systems
1 Introduction
2 Hitchin Integrable Systems
3 Simpson Integrable Systems
4 Spectral Cover via Projective Duality
References
Peter-Weyl Bases, Preferred Deformations, and Schur-Weyl Duality
1 Introduction
2 Deformations
2.1 Formal Deformations
2.2 Equivalence and Preferred Presentations
2.3 Standard Deformation of U(g)
2.4 Standard Deformation of O(Mn)
3 Peter-Weyl Bases and Preferred Deformations
3.1 The Peter-Weyl Basis
3.2 Comultiplication
3.3 Multiplication (Abstract)
3.4 3j Symbols
3.5 Structure Constants for Multiplication
3.6 Preferred Presentation of O.12em.1emdotteddotteddotted.76dotted.6h(G)
3.7 Preferred Deformation of U(g)
3.8 Non-simply connected Groups and Matrix Algebras
4 Relation to Schur-Weyl Duality
4.1 General Categorical Discussion
4.2 Using Vn, Undeformed
4.3 Using Vn, Deformed
4.4 Preferred Presentation
4.5 Comparing with Previous Work
4.6 Deriving the R-Matrix Relations in Oq(M2)
References
Quantum Periodicity and Kirillov–Reshetikhin Modules
1 Introduction
2 Periodicity and Quantum Periodicity
2.1 Periodicity
2.2 Quantum Periodicity
3 Finite-Dimensional Representations of Quantum Affine Algebras
3.1 Quantum Affine Algebras
3.2 Finite-Dimensional Representations
3.3 Quantum Grothendieck Ring
4 Relations in the Grothendieck Ring
4.1 Original T-Systems
4.2 Horizontal T-Systems
5 Proof of Periodicity
References
A Note on the E-Polynomials of a Stratification of the Hilbert Scheme of Points
1 Introduction
2 The Refined Hilbert Schemes
2.1 Definition and Basic Properties
2.2 The Calculation of the E-Polynomial
3 Proof of Theorem 2
4 The E-Polynomial of the Refined Strata H[n]m
5 Euler Characteristics of the Refined Strata
Appendix 1: Properties of the E-Polynomial
Appendix 2: Tables of E-Polynomials of the Refined Strata
References
Galois Action on VOA Gauge Anomalies
1 Gauge Anomalies for Associative Algebras
1.1 Azumaya Algebras
1.2 Physical Interpretation
1.3 Construction of Gauge Anomalies
1.4 Gauging
1.5 Galois Action and the Cohomological Brauer Group
2 Gauge Anomalies for VOAs
2.1 Holomorphic VOAs
2.2 Physical Interpretation
2.3 Construction of Gauge Anomalies
2.4 Gauging
2.5 Linear Algebraic Description
3 Main Result
3.1 Recollection on the Cyclotomic Galois Group
3.2 Nonanomalous Actions
3.3 A Construction of Evans and Gannon
3.4 Completion of the Proof
4 Questions and Conjectures
4.1 Moonshine for Every Group
4.2 A Vertex Brauer Group
4.3 K-Theoretic Interpretation of Anomalies
4.4 Galois and Grothendieck–Teichmuller Groups and Modular Data
4.5 Cyclotomicity of MTCs
References
Heisenberg-Picture Quantum Field Theory
1 Introduction and Motivation
2 Modulation
3 The Point of Pointings
4 Non-affine Field Theory
5 Extended Affine Field Theory
6 Extended Non-Affine Field Theory
7 A (Non-)dualizability Result
8 From Factorization Algebra to Heisenberg-Picture Field Theory
9 From Skein Theory to Heisenberg-Picture Field Theory
References
Irreducibility of the Wysiwyg Representations of Thompson's Groups
1 Introduction
2 Definitions
3 The Wysiwyg Representations
4 The Main Theorem
5 Improvements
References
Invariants of Long Knots
1 Introduction
2 Long Knots
3 Invariants of Long Knots from Rigid R-Matrices
4 Rigid R-Matrices from Racks
4.1 Racks Associated with Pointed Groups
4.2 An Extended Heisenberg Group
5 Invariants from Hopf Algebras
5.1 A Hopf Algebra Associated with a Two-Dimensional Lie Group
References
Rigged Configurations and Unimodality
1 Introduction
1.1 A Bit of History
1.2 Introduction
2 Basic Notation and Definitions
2.1 Polynomials Associated with Configurations of Type (λ,R)
2.2 Formulas for Unimodality Index
3 Main Results
3.1 Generalized Catalan Polynomials
3.2 Some Remarks on Strict Unimodality
3.3 Generalized q-Narayana and Carlitz–Riordan Polynomials
4 Internal Product of Schur Functions
4.1 Liskova Polynomials
4.2 Two-Variable Liskova Polynomials Lαβμ(q,t )
5 Generalized Exponents and Mixed Tensor Representations
6 Rigged Configurations: A Brief Review
6.1 Example
References
Turning Point Processes in Plane Partitions with Periodic Weights of Arbitrary Period
1 Introduction
2 Notation and Description of Main Results
3 The Correlation Kernel and Its Leading Asymptotics
3.1 The Asymptotically Leading Term in the Integral for the Correlation Kernel
4 The Point Processes in the Bulk Near the Edge and at the Turning Points
4.1 The Possible Locations of Double Real Critical Points
4.2 Computing the Double Real Critical Points
4.3 The Number of Real and Complex Critical Points
4.4 Deformation of Contours and the Limiting Correlation Kernel in the Bulk
4.5 Deformation of Contours and the Limiting Correlation Kernel Near Turning Points
4.6 The GUE-Corners Process
4.7 The Frozen Regions Separating the Turning Points
5 Point Processes at the First-Order Phase Transition
5.1 The Double Real Critical Points
5.2 The Number of Critical Points
References
The Skein Category of the Annulus
1 Introduction
2 The Category of Tangle Diagrams
3 The Skein Category of the Annulus
4 Equivalence with the Affine Temperley-Lieb Category
5 The Extended Affine Temperley-Lieb Algebra
6 The Arc Insertion Functor
7 Towers of Extended Affine Temperley-Lieb Algebra Modules
8 The Link Pattern Tower
9 The Link Pattern Tower and Fusion
Appendix: Relation to Affine Hecke Algebras and Affine Braid Groups, and type B Presentations
References
Tensor Product of the Fock Representation with Its Dual and the Deligne Category
1 Introduction
2 The Category RepGL(m|n) and DS Functors
2.1 Translation Functors
2.2 DS-functor
3 The Category Vt, Translation Functors, and Categorification
3.1 The Deligne Category Dt
3.2 The Abelian Envelope of Dt
3.3 Objects of Vt
3.4 Translation Functors and Categorical Action of sl(∞)
4 Proof of the Main Theorem
5 Blocks in Vt and Dimensions of Tilting and Standard Objects
References
Exact Density Matrix for Quantum Group Invariant Sector of XXZ Model
1 Introduction
2 Matsubara Expectation Values
3 Quantum Group Invariant Operators
3.1 Generalities
3.2 Invariant Operators
4 Procedure of Computation
5 The Case of Unbroken Quantum Group Symmetry
5.1 Relation to SOS/RSOS
6 Numerical Results. Zero Temperature
7 Temperature
References
Loops in Surfaces and Star-Fillings
1 Introduction
2 Quasi-Surfaces
2.1 Basics
2.2 Loops in X
2.3 Local Moves
3 Homological Intersection Forms
3.1 The Intersection Form of (X, ω)
3.2 The Intersection Form of X
3.3 Proof of Theorem 3.2
3.4 Computation of •X
4 The Intersection Brackets
4.1 The Brackets
4.2 The Pairing μk
4.3 Proof of Theorem 4.2
4.4 Computation of [-,-]X
4.5 Remarks
5 The Cobrackets
5.1 The Cobracket νX,ω
5.2 The Cobracket νX
5.3 Computation of νX
5.4 Examples
6 Transformations of Quasi-Surfaces
6.1 The Transformation D
6.2 The Transformation C
7 Stars in Quasi-Surfaces
7.1 Stars in X
7.2 Star-Fillings of X
8 Proof of Theorems 1.1–1.4 and the Case of Closed Surfaces
8.1 Proof of Theorems 1.1–1.4
8.2 The Case of Closed Surfaces
8.3 Example
References