Quantum groups have been studied intensively in mathematics and have found many valuable applications in theoretical and mathematical physics since their discovery in the mid-1980s. Roughly speaking, there are two prototype examples of quantum groups, denoted by Uq and Aq. The former is a deformation of the universal enveloping algebra of a Kac–Moody Lie algebra, whereas the latter is a deformation of the coordinate ring of a Lie group. Although they are dual to each other in principle, most of the applications so far are based on Uq, and the main targets are solvable lattice models in 2-dimensions or quantum field theories in 1+1 dimensions. This book aims to present a unique approach to 3-dimensional integrability based on Aq. It starts from the tetrahedron equation, a 3-dimensional analogue of the Yang–Baxter equation, and its solution due to work by Kapranov–Voevodsky (1994). Then, it guides readers to its variety of generalizations, relations to quantum groups, and applications. They include a connection to the Poincaré–Birkhoff–Witt basis of a unipotent part of Uq, reductions to the solutions of the Yang–Baxter equation, reflection equation, G2 reflection equation, matrix product constructions of quantum R matrices and reflection K matrices, stationary measures of multi-species simple-exclusion processes, etc. These contents of the book are quite distinct from conventional approaches and will stimulate and enrich the theories of quantum groups and integrable systems.
Author(s): Atsuo Kuniba
Series: Theoretical and Mathematical Physics
Publisher: Springer
Year: 2022
Language: English
Pages: 329
City: Singapore
Preface
Contents
1 Introduction
1.1 Quantum Integrability in Two Dimensions
1.2 Quantization: Introducing the Third Dimension
1.3 Quantized Coordinate Ring
1.4 Compatibility: Tetrahedron, 3D Reflection and upper F 4F4 Equations
1.5 Feedback to 2D
1.6 Layout of the Book
2 Tetrahedron Equation
2.1 3D MathID3R
2.2 Tetrahedron Equation of Type MathID35RRRR=RRRR
2.3 3D MathID71L
2.4 Tetrahedron Equation of Type MathID97RLLL = LLLR
2.5 Quantized Yang–Baxter Equation
2.6 Tetrahedron Equation of Type MathID140MMLL=LLMM
2.7 Bibliographical Notes and Comments
3 3D upper RR From Quantized Coordinate Ring of Type A
3.1 Quantized Coordinate Ring upper A Subscript q Baseline left parenthesis upper A Subscript n minus 1 Baseline right parenthesisAq(An-1)
3.2 Representation Theory
3.3 Intertwiner for Cubic Coxeter Relation
3.4 Explicit Formula for 3D upper RR
3.5 Solution to the Tetrahedron Equations
3.5.1 upper R upper R upper R upper R equals upper R upper R upper R upper RRRRR=RRRR Type
3.5.2 upper R upper L upper L upper L equals upper L upper L upper L upper RRLLL=LLLR Type
3.5.3 upper M upper M upper L upper L equals upper L upper L upper M upper MMMLL=LLMM Type
3.6 Further Aspects of 3D upper RR
3.6.1 Boundary Vector
3.6.2 Combinatorial and Birational Counterparts
3.6.3 Bilinearization and Geometric Interpretation
3.7 Bibliographical Notes and Comments
4 3D Reflection Equation and Quantized Reflection Equation
4.1 Introduction
4.2 3D upper KK
4.3 3D Reflection Equation
4.4 Quantized Reflection Equation
4.5 Bibliographical Notes and Comments
5 3D MathID2K From Quantized Coordinate Ring of Type C
5.1 Quantized Coordinate Ring MathID6Aq(Cn)
5.2 Fundamental Representations
5.3 Interwtiners for Quadratic and Cubic Coxeter Relations
5.4 Intertwiner for Quartic Coxeter Relation
5.5 Explicit Formula for 3D MathID187K
5.6 Solution to the 3D Reflection Equation
5.7 Solution to the Quantized Reflection Equation
5.8 Further Aspects of 3D MathID361K
5.8.1 Boundary Vector
5.8.2 Combinatorial and Birational Counterparts
5.9 Bibliographical Notes and Comments
6 3D upper KK From Quantized Coordinate Ring of Type B
6.1 Quantized Coordinate Ring upper A Subscript q Baseline left parenthesis upper B Subscript n Baseline right parenthesisAq(Bn)
6.2 Fundamental Representations
6.3 Intertwiners
6.4 3D Reflection Equation
6.5 Combinatorial and Birational Counterparts
6.6 Proof of Proposition 6.5
6.6.1 Matrix Product Formula of the Structure Function
6.6.2 upper R upper T upper TRTT Relation
6.6.3 rho upper T upper TρTT Relations
6.7 Bibliographical Notes and Comments
7 Intertwiners for Quantized Coordinate Ring upper A Subscript q Baseline left parenthesis upper F 4 right parenthesisAq(F4)
7.1 Fundamental Representations
7.2 Intertwiners
7.3 upper F 4F4 Analogue of the Tetrahedron/3D Reflection Equations
7.4 Reduction to 3D Reflection Equations
7.5 Bibliographical Notes and Comments
8 Intertwiner for Quantized Coordinate Ring upper A Subscript q Baseline left parenthesis upper G 2 right parenthesisAq(G2)
8.1 Introduction
8.2 Quantized Coordinate Ring upper A Subscript q Baseline left parenthesis upper G 2 right parenthesisAq(G2)
8.3 Fundamental Representations
8.4 Intertwiner
8.5 Quantized upper G 2G2 Reflection Equation
8.5.1 3D upper LL
8.5.2 Quantized upper G 2G2 Scattering Operator upper JJ
8.5.3 Quantized upper G 2G2 Reflection Equation
8.6 Further Aspects of upper FF
8.6.1 Boundary Vector
8.6.2 Combinatorial and Birational Counterparts
8.7 Data on Relevant Quantum upper RR Matrix
8.8 Bibliographical Notes and Comments
9 Comments on Tetrahedron-Type Equation for Non-crystallographic Coxeter Groups
9.1 Finite Coxeter Groups
9.2 Tetrahedron-Type Equation for the Coxeter Group ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper H 3) /StPNE pdfmark [/StBMC pdfmarkH3ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark
9.3 Discussion on the Quintic Coxeter Relation
10 Connection to PBW Bases of Nilpotent Subalgebra of upper U Subscript qUq
10.1 Quantized Universal Enveloping Algebra upper U Subscript q Baseline left parenthesis German g right parenthesisUq(mathfrakg)
10.1.1 Definition
10.1.2 PBW Basis
10.2 Quantized Coordinate Ring upper A Subscript q Baseline left parenthesis German g right parenthesisAq(mathfrakg)
10.2.1 Definition
10.2.2 Right Quotient Ring upper A Subscript q Baseline left parenthesis German g right parenthesis Subscript script upper SAq(mathfrakg)S
10.3 Main Theorem
10.3.1 Definitions of gamma Subscript upper B Superscript upper AγAB and normal upper Phi Subscript upper B Superscript upper AΦAB
10.3.2 Proof of Theorem 10.6 for Rank 2 Cases
10.4 Proof of Proposition 10.7
10.4.1 Explicit Formulas for upper A 2A2
10.4.2 Explicit Formulas for upper C 2C2
10.4.3 Explicit Formulas for upper G 2G2
10.5 Tetrahedron and 3D Reflection Equations from PBW Bases
10.6 chiχ-Invariants
10.7 Bibliographical Notes and Comments
11 Trace Reductions of upper R upper L upper L upper L equals upper L upper L upper L upper RRLLL=LLLR
11.1 Introduction
11.2 Trace Reduction Over the Third Component of upper LL
11.3 Trace Reduction Over the First Component of upper LL
11.4 Trace Reduction Over the Second Component of upper LL
11.5 Identification with Quantum upper RR Matrices of upper A Subscript n minus 1 Superscript left parenthesis 1 right parenthesisA(1)n-1
11.5.1 upper S Superscript trace Super Subscript 3 Baseline left parenthesis z right parenthesisStr3(z)
11.5.2 upper S Superscript trace Super Subscript 1 Baseline left parenthesis z right parenthesisStr1(z)
11.5.3 upper S Superscript trace Super Subscript 2 Baseline left parenthesis z right parenthesisStr2(z)
11.6 Commuting Layer Transfer Matrices and Duality
11.7 Bibliographical Notes and Comments
12 Boundary Vector Reductions of upper R upper L upper L upper L equals upper L upper L upper L upper RRLLL=LLLR
12.1 Boundary Vector Reductions
12.2 Identification with Quantum upper RR Matrices of upper B Subscript n Superscript left parenthesis 1 right parenthesis Baseline comma upper D Subscript n Superscript left parenthesis 1 right parenthesis Baseline comma upper D Subscript n plus 1 Superscript left parenthesis 2 right parenthesisB(1)n, D(1)n, D(2)n+1
12.2.1 Quantum Affine Algebra upper U Subscript p Baseline left parenthesis German g Superscript r comma r prime Baseline right parenthesisUp(mathfrakgr,r')
12.2.2 Spin Representations of upper U Subscript p Baseline left parenthesis German g Superscript r comma r prime Baseline right parenthesisUp(mathfrakgr,r')
12.2.3 upper S Superscript r comma r Super Superscript prime Superscript Baseline left parenthesis z right parenthesisSr,r'(z) as Quantum upper RR Matrices for Spin Representations
12.3 Commuting Layer Transfer Matrix
12.4 Examples of upper S Superscript 1 comma 1 Baseline left parenthesis z right parenthesis comma upper S Superscript 2 comma 1 Baseline left parenthesis z right parenthesis comma upper S Superscript 2 comma 2 Baseline left parenthesis z right parenthesisS1,1(z), S2,1(z), S2,2(z) for n equals 2n=2
12.5 Bibliographical Notes and Comments
13 Trace Reductions of upper R upper R upper R upper R equals upper R upper R upper R upper RRRRR=RRRR
13.1 Preliminaries
13.2 Trace Reduction Over the Third Component of upper RR
13.3 Trace Reduction Over the First Component of upper RR
13.4 Trace Reduction Over the Second Component of upper RR
13.5 Explicit Formulas of upper R Superscript trace Super Subscript 1 Superscript Baseline left parenthesis z right parenthesis comma upper R Superscript trace Super Subscript 2 Superscript Baseline left parenthesis z right parenthesis comma upper R Superscript trace Super Subscript 3 Superscript Baseline left parenthesis z right parenthesisRtr1(z), Rtr2(z), Rtr3(z)
13.5.1 Function upper A left parenthesis z right parenthesis Subscript bold i bold j Superscript bold a bold bA(z)a bij
13.5.2 upper A left parenthesis z right parenthesis Subscript bold i bold j Superscript bold a bold bA(z)a bij as Elements of upper R Superscript trace Super Subscript 1 Superscript Baseline left parenthesis z right parenthesis comma upper R Superscript trace Super Subscript 2 Superscript Baseline left parenthesis z right parenthesisRtr1(z),Rtr2(z) and upper R Superscript trace Super Subscript 3 Baseline left parenthesis z right parenthesisRtr3(z)
13.5.3 Proof of Theorem 13.3
13.6 Identification with Quantum upper RR Matrices of upper A Subscript n minus 1 Superscript left parenthesis 1 right parenthesisA(1)n-1
13.6.1 upper R Superscript trace Super Subscript 3 Baseline left parenthesis z right parenthesisRtr3(z)
13.6.2 upper R Superscript trace Super Subscript 1 Baseline left parenthesis z right parenthesisRtr1(z)
13.6.3 upper R Superscript trace Super Subscript 2 Baseline left parenthesis z right parenthesisRtr2(z)
13.7 Stochastic upper RR Matrix
13.8 Commuting Layer Transfer Matrices and Duality
13.9 Geometric upper RR From Trace Reductions of Birational 3D upper RR
13.10 Bibliographical Notes and Comments
14 Boundary Vector Reductions of upper R upper R upper R upper R equals upper R upper R upper R upper RRRRR=RRRR
14.1 Boundary Vector Reductions
14.1.1 nn-Concatenation of the Tetrahedron Equation
14.1.2 Boundary Vector Reductions
14.2 Identification with Quantum upper RR Matrices of upper A Subscript 2 n Superscript left parenthesis 2 right parenthesis Baseline comma upper C Subscript n Superscript left parenthesis 1 right parenthesis Baseline comma upper D Subscript n plus 1 Superscript left parenthesis 2 right parenthesisA(2)2n, C(1)n, D(2)n+1
14.2.1 Quantum Affine Algebra upper U Subscript q Baseline left parenthesis German g Superscript r comma r prime Baseline right parenthesisUq(mathfrakgr,r').
14.2.2 qq-Oscillator Representations
14.2.3 Quantum Group Symmetry
14.3 Bibliographical Notes and Comments
15 Trace Reduction of left parenthesis upper L upper G upper L upper G right parenthesis upper K equals upper K left parenthesis upper G upper L upper G upper L right parenthesis(LGLG)K= K(GLGL)
15.1 Introduction
15.2 Concatenation of Quantized Reflection Equation
15.3 Trace Reduction
15.4 Characterization as the Intertwiner of the Onsager Coideal
15.4.1 Generalized pp-Onsager Algebra upper O Subscript p Baseline left parenthesis upper A Subscript n minus 1 Superscript left parenthesis 1 right parenthesis Baseline right parenthesisOp(A(1)n-1)
15.4.2 upper K Superscript trace Baseline left parenthesis z right parenthesisKtr(z) as the Intertwiner of Onsager Coideal
15.4.3 Reflection Equation From Onsager Coideal
15.5 Further Properties of upper K Superscript trace Baseline left parenthesis z right parenthesisKtr(z)
15.5.1 Commutativity
15.5.2 upper K Superscript trace Baseline left parenthesis z right parenthesisKtr(z) as a Symmetry of XXZ-Type Spin Chain
15.6 Bibliographical Notes and Comments
16 Boundary Vector Reductions of left parenthesis upper L upper G upper L upper G right parenthesis upper K equals upper K left parenthesis upper G upper L upper G upper L right parenthesis(LGLG)K= K(GLGL)
16.1 Preliminaries
16.2 Boundary Vector Reduction
16.3 Characterization as the Intertwiner of the Onsager Coideal
16.3.1 Generalized pp-Onsager Algebra upper O Subscript p Baseline left parenthesis German g Superscript r comma r prime Baseline right parenthesisOp(mathfrakgr,r')
16.3.2 upper K Superscript k comma k Super Superscript prime Superscript Baseline left parenthesis z right parenthesisKk,k'(z) as the Intertwiner of Onsager Coideal
16.4 Bibliographical Notes and Comments
17 Reductions of Quantized MathID2G2 Reflection Equation
17.1 Introduction
17.2 The MathID38G2 Reflection Equation
17.3 Quantized MathID135G2 Reflection Equation
17.4 Reduction of the Quantized MathID158G2 Reflection Equation
17.4.1 Concatenation of Quantized MathID161G2 Reflection Equation
17.4.2 Trace Reduction
17.4.3 Boundary Vector Reduction
17.5 Properties of MathID219Xtr(z) and MathID220Xbv(z)
17.6 Bibliographical Notes and Comments
18 Application to Multispecies TASEP
18.1 Introduction
18.2 nn-TASEP
18.2.1 Definition of nn-TASEP
18.2.2 Stationary States
18.2.3 Matrix Product Formula
18.2.4 Matrix Product Operator upper X Subscript i Baseline left parenthesis z right parenthesisXi(z)
18.3 3D upper L comma upper ML, M Operators and the Tetrahedron Equation
18.4 Layer Transfer Matrices
18.4.1 Layer Transfer Matrices with Mixed Boundary Condition
18.4.2 Commutativity
18.4.3 Bilinear Identities of Layer Transfer Matrices
18.5 Proof of Theorem 18.5
18.6 Bibliographical Notes and Comments
Appendix References
Index
