Quantum Groups in Three-Dimensional Integrability

Beschreibung

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 U_q and A_q. 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 U_q, 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 A_q. It starts from the tetrahedron equation, a 3-dimensional analogue of the Yang–Baxter equation, and its solution due to work byKapranov–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 U_q, 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.

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 U_q and A_q. 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 U_q, 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 A_q. 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 U_q, 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.


Presents quantized coordinate ring as a main player dual to what is usually meant by quantum group in physics literature Illustrates quantization of the conventional Yang–Baxter and reflection equations, related to 3D integrability Leads to matrix product formulas for R and K matrices having intriguing applications

Autor*in

Atsuo Kuniba

Themen in »Quantum Groups in Three-Dimensional Integrability«

Quantized Coordinate Ring Tetrahedron /3D Reflection Equations PBW Basis of Positive Part of Quantized Enveloping Algebra Q-Boson Matrix Product Formula for R and K Matrices Generalized Onsager Algebra/Coideal

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“This book is well written, with numerous examples, bibliographical information, references, and pictures. Essentially, all of the computations are based around algebraic manipulations, and generally require no deep knowledge of representation theory, algebraic geometry, or integrable systems. … this is a very good book to move into the third dimension for studying integrable systems.” (Travis Scrimshaw, Mathematical Reviews, February, 2024)

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