Quantum Groups in Three-Dimensional Integrability

Atsuo Kuniba (Autor)

Buch | Hardcover

331 Seiten

2022 | 2022 ed.
Springer Verlag, Singapore
978-981-19-3261-8 (ISBN)

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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.

Introduction.- Tetrahedron equation.- 3D R from quantized coordinate ring of type A.- 3D reﬂection equation and quantized reﬂection equation.- 3D K from quantized coordinate ring of type C.- 3D K from quantized coordinate ring of type B.- Intertwiners for quantized coordinate ring Aq (F4).- Intertwiner for quantized coordinate ring Aq (G2).- Comments on tetrahedron-type equation for non-crystallographic Coxeter groups.- Connection to PBW bases of nilpotent subalgebra of Uq.- Trace reductions of RLLL = LLLR.- Boundary vector reductions of RLLL = LLLR.- Trace reductions of RRRR = RRRR.- Boundary vector reductions of RRRR = RRRR.- Boundary vector reductions of (LGLG)K = K(GLGL).- Reductions of quantized G2 reﬂection equation.- Application to multispecies TASEP.

Erscheinungsdatum	28.09.2022
Reihe/Serie	Theoretical and Mathematical Physics
Zusatzinfo	32 Illustrations, color; 64 Illustrations, black and white
Verlagsort	Singapore
Sprache	englisch
Maße	155 x 235 mm
Themenwelt	Mathematik / Informatik ► Mathematik
	Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
	Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
ISBN-10	981-19-3261-1 / 9811932611
ISBN-13	978-981-19-3261-8 / 9789811932618
Zustand	Neuware