Quantum Groups in Three-Dimensional Integrability (eBook)

Atsuo Kuniba (Autor)

eBook Download: PDF

2022 | 1st ed. 2022
XI, 331 Seiten
Springer Nature Singapore (Verlag)
978-981-19-3262-5 (ISBN)

Infos

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.

---

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 Poincare-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.

Produkt-Details

• Erscheint lt. Verlag: 25.9.2022
• Reihe/Serie: Theoretical and Mathematical Physics
• Reihe/Serie: Theoretical and Mathematical Physics
• Zusatzinfo: XI, 331 p. 96 illus., 32 illus. in color.
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Schlagworte: Generalized Onsager Algebra/Coideal • PBW Basis of Positive Part of Quantized Enveloping Algebra • Q-Boson Matrix Product Formula for R and K Matrices • Quantized Coordinate Ring • Tetrahedron /3D Reflection Equations
• ISBN-10: 981-19-3262-X / 981193262X
• ISBN-13: 978-981-19-3262-5 / 9789811932625