quantum transformation groupoid

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 $ 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

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

:

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

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

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$ .



Contributors to this entry (in most recent order):

As of this snapshot date, this entry was owned by bci1.