Abstract
It is shown that, on the one hand, quantum moment maps give rise to examples for the operator-theoretic approach to invariant integration theory developed by K.-D. Kürsten and the second author, and that, on the other hand, the operator-theoretic approach to invariant integration theory is more general since it also applies to examples without a well-defined quantum moment map.
Highlights
A noncommutative analogue of an group action on a topological space is described by the action of a Hopf algebra on a noncommutative function algebra
The noncommutative function algebra is generated by a finite set of generators which are considered as coordinate functions on the quantum space
As in the classical case, one does not expect that polynomials in the coordinate functions on locally compact quantum spaces are integrable
Summary
A noncommutative analogue of an (infinitesimal) group action on a topological space is described by the action of a Hopf algebra on a noncommutative function algebra. Any joint representation of U and A on the same Hilbert space allows one to equip A with a U-action, given by the formulas of the adjoint action, provided that A is invariant under these algebraic expressions [6]. This will be automatically the case if there is a. We demonstrate that the operator-theoretic approach to invariant integration theory is more general, since it applies to cases where the operators describing the action do not satisfy the commutation relations of U and do not define a quantum moment map
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