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

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

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