Integrable actions and quantum subgroups

Abstract

We study homomorphisms of locally compact quantum groups from the point of view of integrability of the associated action. For a given homomorphism of quantum groups Π : H → G we introduce quantum groups H/ker Π and im Π corresponding to the classical quotient by kernel and closure of image. We show that if the action of H on G associated to Π is integrable then H/ker Π ∼ = im Π and characterize such Π. As a particular case we consider an injective continuous homomorphism Π : H → G between locally compact groups H and G. Then Π yields an integrable action of H on L ∞ (G) if and only if its image is closed and Π is a homeomorphism of H onto im Π.We also give characterizations of open quantum subgroups and of compact quantum subgroups in terms of integrability and show that a closed quantum subgroup always gives rise to an integrable action. Moreover we prove that quantum subgroups closed in the sense of Woronowicz whose associated homomorphism of quantum groups yields an integrable action are closed in the sense of Vaes.2010 Mathematics Subject Classification. Primary: 46L89 Secondary: 46L85, 46L52. Key words and phrases. Locally compact quantum group, quantum subgroup, homomorphism of quantum groups, integrable action.

Fundamental isomorphism theorems for quantum groups

“…In what follows we shall provide a number of descriptions of H ker Π and formulate the condition which yields the existence of ker Π entering the exact sequence (2.6). The von Neumann algebra L ∞ (H ker Π) is defined as (see [16,Definition 4.4]…”

“…∎ A morphism Π ∶ H → G is assigned with the dual morphismΠ ∶Ĝ →Ĥ which in terms of bicharacter is given byV = σ(V ) * . The locally compact quantum groupĜ kerΠ will be denoted by imΠ (see [16,Definition 4.3]). In particular, using (2.7) we can see that The next lemma will be needed further.…”

“…The preceding discussion explains why this will do. In general, we say that a morphism Π ∶ P → Q of locally compact quantum groups has trivial kernel if the quotient quantum group P → P kerΠ of [16,Definition 4.4] is an isomorphism. Let us recal that Π ∶ P → Q induces a morphism Π 1 ∶ P ker Π → Q which has trivial kernel.…”

“…), the techniques used in practice and the attendant technical difficulties are often specific to either the analytic or the algebraic framework. For instance, as we will see below, for locally compact quantum groups one obstruction to obtaining the types of results we seek will be the lack of integrability (in the sense of [16]) for a quantum group action on a non-commutative space.…”

Quantum relative modular functions

“…The last equality in (2-20) is nothing but (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14), whereas the first equality will follow once we have (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21); it thus remains to prove the latter. For that purpose, note that for…”