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.

Definition 1.1 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

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

$\displaystyle \mathbb{H} = L^2(A) \otimes L^2(M)$

$\displaystyle \beta(x) = x \otimes 1,$

$\displaystyle \hat \beta(x) = (J_A \otimes J_M)\alpha(x^*)(J_A \otimes J_M),$

with $\alpha$ 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 $A$ .

"quantum transformation groupoid" is owned by bci1.
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