1 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), Glc and also
let Xlc be a locally compact space underlying Glc . If A and M are von Neumann algebras
and (M, Δ) is a (von Neumann) locally compact group, then one can define the following
representations of A on a Hilbert space
:
with α being the left action of (M, Δ) on ℍ.
A quantum transformation groupoid 𝔾T is defined by the α left action of (M, Δ) on ℍ which
has the above representations of A.
