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Noncommutative Geometry on enveloping algebras and its applications to Physics
(Université de Valeciennes)
Amphithéâtre Léon Motchane (IHES)
Amphithéâtre Léon Motchane
Le Bois Marie
35, route de Chartres
The central problem of Noncommutative Geometry is constructing differential calculus on a given noncommutative algebra. Some known approaches to this problem will be mentioned in my talk.
Also, I shall exhibit a new approach to constructing such a calculus on the enveloping algebras of Lie algebras gl(n) and their super-analogs. This approach is based on a new form of the Leibniz rule. As a result, the corresponding differential algebra can be treated as a quantization (deformation) of its commutative counterpart, namely, the differential algebra on the symmetric algebra of a given Lie algebra gl(n).
The role of braided algebras (i.e., those related to the corresponding quantum groups) in constructing this calculus will be exhibited. Applications to quantization of some dynamical models by means of so-called "quantum spherical coordinates" will be also exhibited.