Quantum Groups in Three-Dimensional Integrability

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