This tag is for the questions relating to Heisenberg group (or Weyl-Heisenberg group) which is a Lie group integrating a Heisenberg Lie algebra. It is another illustration of its perception as an extraneous object: physicists call it by the name of a mathematician, and mathematicians by the name of a physicists.
A Heisenberg group is a Lie group whose Lie algebra is a Heisenberg Lie algebra.
Heisenberg group is also known as the Weyl (or Heisenberg–Weyl) group.
The Heisenberg group historically originates in and still has its strongest ties to quantum physics: there it is a group of unitary operators acting on the space of states induced from those observables on a linear phase space – a symplectic vector space – which are given by linear or by constant functions. So any Heisenberg group is a subgroup of a group of observables in certain simple examples of quantum mechanical systems.
Heisenberg group reveals itself as an important factor in many apparently diverse topics like,
- Representation Theory of Nilpotent Lie Groups
- Foundations of Abelian Harmonic Analysis
- Moduli of Abelian Varieties
- Structure Theory of Finite Groups
- Theory of Partial Differential Equations
- Quantum Mechanics
- Homological Algebra
- Ergodic Theory
- Representation Theory of Reductive Algebraic groups
- Classical Invariant Theory
This list could easily be lengthened both by adding new topics and making these more specific, for sometimes the applications are multiple.
References:
https://en.wikipedia.org/wiki/Heisenberg_group
https://www.univie.ac.at/nuhag-php/bibtex/open_files/1389_fr750001.pdf
