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.
