# PlanetPhysics/Quantum Transformation Groupoid

\newcommand{\sqdiagram}[9]{**Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}}**
}

## Quantum transformation groupoidEdit

This is a quantum analog construction of the classical transformation group
construction *via* the action of a group on a state (or phase) space.

Let us a consider a locally compact quantum group (L-CQG), and also let be a locally compact space underlying . If and are von Neumann algebras and is a (von Neumann) locally compact group, then one can define the following representations of on a Hilbert space :

with being the left action of on .

A *quantum transformation groupoid* is defined by the left action of on which has the above representations of .