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 $$A$$.