# PlanetPhysics/Quantum Transformation Groupoid

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## Quantum transformation groupoid

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), ${\displaystyle G_{lc}}$  and also let ${\displaystyle X_{lc}}$  be a locally compact space underlying ${\displaystyle G_{lc}}$  . If ${\displaystyle A}$  and ${\displaystyle M}$  are von Neumann algebras and ${\displaystyle (M,\Delta )}$  is a (von Neumann) locally compact group, then one can define the following representations of ${\displaystyle A}$  on a Hilbert space ${\displaystyle \mathbb {H} =L^{2}(A)\otimes L^{2}(M)}$ :

${\displaystyle \beta (x)=x\otimes 1,}$  ${\displaystyle {\hat {\beta }}(x)=(J_{A}\otimes J_{M})\alpha (x^{*})(J_{A}\otimes J_{M}),}$  with ${\displaystyle \alpha }$  being the left action of ${\displaystyle (M,\Delta )}$  on ${\displaystyle \mathbb {H} }$ .

A quantum transformation groupoid ${\displaystyle \mathbb {G} _{T}}$  is defined by the ${\displaystyle \alpha }$  left action of ${\displaystyle (M,\Delta )}$  on ${\displaystyle \mathbb {H} }$  which has the above representations of ${\displaystyle A}$ .