Invariant Differential Operators. Volume 2: Quantum Groups

Main subject categories: • Group theory and generalizations • Rings and algebras • Quantum Theory • Mathematical Physics • and more[From the preface]This is volume 2 of our set on invariant differential operators. In volume 1 we presented our canonical procedure for the construction of invariant differential operators and showed its application to the objects of the initial domain – noncompact semisimple Lie algebras and groups.In volume 2 we show the application of our procedure to quantum groups. Similarly to the setting of volume 1 the main actors are in duality. Just as Lie algebras and Lie groups are in duality here the dual objects are the main two manifestations of quantum groups: quantum algebras and matrix quantum groups. Actually, quantum algebras typically are deformations of the universal enveloping algebras of semisimple Lie algebras. Analogously, matrix quantum groups typically are deformations of spaces of functions over semisimple Lie groups. With applications in quantum field theory, general relativity and elementary particle physics, this three-volume work studies the invariance of differential operators under Lie algebras, quantum groups and superalgebras. This second volume covers quantum groups in their two main manifestations: quantum algebras and matrix quantum groups. The exposition covers both the general aspects of these and a great variety of concrete explicitly presented examples. The invariant q-difference operators are introduced mainly using representations of quantum algebras on their dual matrix quantum groups as carrier spaces. This is the first book that covers the title matter applied to quantum groups. Contents Quantum Groups and Quantum Algebras Highest-Weight Modules over Quantum Algebras Positive-Energy Representations of Noncompact Quantum Algebras Duality for Quantum Groups Invariant q-Difference Operators Invariant q-Difference Operators Related to GLq(n) q-Maxwell Equations Hierarchies With Applications In Quantum Field Theory, Elementary Particle Physics And General Relativity, This Two-volume Work Studies Invariance Of Differential Operators Under Lie Algebras, Quantum Groups, Superalgebras Including Infinite-dimensional Cases, Schrödinger Algebras, Applications To Holography-- "With applications in quantum field theory, elementary particle physics and general relativity, this four-volume work studies invariance of differential operators under Lie algebras, quantum groups, and superalgebras"-- Provided by publisher