Seminar talk, 18 May 2022

From Geometry of Differential Equations
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Speaker: Andrei Smilga

Title: Noncommutative quantum mechanical systems associated with Lie algebras

We consider quantum mechanics on the noncommutative spaces characterized by the commutation relations


where are the structure constants of a Lie algebra. We note that this problem can be reformulated as an ordinary quantum problem in a commuting momentum space. The coordinates are then represented as linear differential operators . Generically, the matrix represents a certain infinite series over the deformation parameter : . The deformed Hamiltonian, describes the motion along the corresponding group manifolds with the characteristic size of order . Their metrics are also expressed into certain infinite series in .

For the algebras and , it has been possible to represent the operators in a simple finite form. A byproduct of our study are new nonstandard formulas for the metrics on and on .