Reading Schrödinger group

Time: 11:00–12:00 on Thursdays of odd-numbered teaching weeks (12 & 26 Oct, 9 & 13 Nov, 7 Dec).
Place: MAGIC room.


1  Motivation

The most prominent examples of three-dimensional Lie groups are: the Heisenberg group ℍ and the group SL2(ℝ). The Heisenberg group initiates the class of nilpotent Lie groups and SL2(ℝ) is the smallest semisimple Lie group. In a sense, those two classes are bearing many opposite properties.

For a wider community, ℍ and SL2(ℝ) are significant for their applications:

The above two lists are partially overlapping and partially complementing. There are deep reasons for this. The mentioned generators of the Lie algebra sl2—the Hamiltonian of the harmonic oscillator and the ladder operators—are quadratic elements of the Heisenberg Lie algebra. In the opposite direction: the group SL2(ℝ) naturally acts by outer automorphisms of the Heisenberg group, thus one can build the semidirect product S=ℍ⋊ SL2(ℝ), which is known as the Schrödinger group.

The Schrödinger group naturally covers any of the above applications of ℍ and SL2(ℝ) and especially efficient in the areas common to both: harmonic analysis, quantum mechanics and number theory. It is enough to mention that:

2  Outline

Our aim is to learn fundamentals of the Schrödinger group in a pedestrian way, which is suitable for young researchers. The initial plan is:

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Last modified: February 18, 2018.
This document was translated from LATEX by HEVEA.