Seminario Q-Math
Enveloping algebras as algebras of bounded Hilbert space operators
Ponente: Wend Werner (Universität Münster - Mathematisches Institut)Fecha: martes 01 de abril de 2025 - 13:00Lugar: Room 2.2.D08, Universidad Carlos III de Madrid
Online: https://eu.bbcollab.com/guest/22f7877a774148a3aee3e398a4a86380 (active on request)
Resumen:
The transition from macro- to microscopic Physics can be carried out in a number of different ways. One of them is based on the deformation of algebras, a topic that is also of great interest within pure mathematics itself. Other so called quantization procedures yield Hilbert space operators very early on; deformation quantization, however, while being quite successful from a purely algebraic point of view, has often difficulties in finding some.
We have a look at the universal enveloping algebra \(U(g)\) of a Lie algebra \(g\). When set up properly, such an algebra can be thought of as being algebraically generated by some physical relevant observables, and it also comes equipped with a very natural deformation into a commutative algebra.
In joint work with Elizabeth Gillaspy we show that U(g) can be represented as an algebra of bounded Hilbert space operators iff the underlying Lie algebra is nilpotent. We additionally try to find a pattern inside the class of all such realizations of \(U(g)\) for a fixed Lie algebra \(g\).
