Quantum Groups in Three-Dimensional Integrability

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 KacMoody 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 YangBaxter equation, and its solution due to work by KapranovVoevodsky (1994). Then, it guides readers to its variety of generalizations, relations to quantum groups, and applications. They include a connection to the PoincaréBirkhoffWitt basis of a unipotent part of U q , reductions to the solutions of the YangBaxter equation, reflection equation, G 2 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 YangBaxter and reflection equations, related to 3D integrability Leads to matrix product formulas for R and K matrices having intriguing applications

Inhalt
Introduction.- Tetrahedron equation.- 3D R from quantized coordinate ring of type A.- 3D reection equation and quantized reection equation.- 3D K from quantized coordinate ring of type C.- 3D K from quantized coordinate ring of type B.- Intertwiners for quantized coordinate ring A q ( F 4 ).- Intertwiner for quantized coordinate ring A q ( G 2 ).- Comments on tetrahedron-type equation for non-crystallographic Coxeter groups.- Connection to PBW bases of nilpotent subalgebra of U q .- 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 G 2 reection equation.- Application to multispecies TASEP.