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 [[../TrivialGroupoid/|group]] on a state (or phase) space.

Let us a consider a [[../LCQG/|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 [[../QuantumOperatorAlgebra4/|von Neumann algebras]] and $(M,\mathrm{\Delta })$ is a (von Neumann) locally compact group, then one can define the following [[../CategoricalGroupRepresentation/|representations]] of $A$ on a [[../NormInducedByInnerProduct/|Hilbert space]] $ℍ=L^2(A)\otimes L^2(M)$:

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

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