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Books > Science & Mathematics > Physics > Quantum physics (quantum mechanics)
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Quantum Groups in Three-Dimensional Integrability (Hardcover, 1st ed. 2022)
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Quantum Groups in Three-Dimensional Integrability (Hardcover, 1st ed. 2022)
Series: Theoretical and Mathematical Physics
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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 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.
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