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OverviewQuantum 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. Full Product DetailsAuthor: Atsuo KunibaPublisher: Springer Verlag, Singapore Imprint: Springer Verlag, Singapore Edition: 1st ed. 2022 Weight: 0.528kg ISBN: 9789811932649ISBN 10: 9811932646 Pages: 331 Publication Date: 27 September 2023 Audience: Professional and scholarly , Professional & Vocational Format: Paperback Publisher's Status: Active Availability: Manufactured on demand  We will order this item for you from a manufactured on demand supplier. Table of ContentsIntroduction.- 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.Reviews“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) Author InformationTab Content 6Author Website:Countries AvailableAll regions |
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