Definition 1.1
Let us a consider a
locally compact quantum group
(L-CQG),

and
also let

be a locally compact space underlying

. If

and

are
von Neumann algebras
and

is a (von Neumann) locally compact group, then one can define the following
representations
of

on a Hilbert space
:
with

being the left action of

on

.
A quantum transformation groupoid
is defined by the
left action of
on
which has the above representations of
.