Induced representation and Frobenius reciprocity for compact quantum groups

We consider compact matrix quantum groups whose $N$-dimensional fundamental representation decomposes into an $(N-1)$-dimensional and a one-dimensional subrepresentation. Even if we know that the compact matrix quantum group associated to this $(N-1)$-dimensional subrepresentation is isomorphic to the given $N$-dimensional one, it is a priori not clear how the intertwiner spaces transform under this isomorphism. In the context of so-called easy and non-easy quantum groups, we are able to define a transformation of linear combinations of partitions and we explicitly describe the transformation of intertwiner spaces. As a side effect, this enables us to produce many new examples of non-easy quantum groups being isomorphic to easy quantum groups as compact quantum groups but not as compact matrix quantum groups.

On square roots of the Haar state on compact quantum groups

Generalizing the notion of continuous Hilbert space representations of compact topological groups we define unitary continuous corepresentations of C*-completions of compact quantum group Hopf algebras on arbitrary Hilbert spaces. It is proved that the unitary continuous corepresentations decompose in finite-dimensional irreducible corepresentations.

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

This is a transcript of a series of eight lectures, 90 min each, held at IMSc Chennai, India from 5–24 January 2015. We give basic definitions, properties and examples of compact quantum groups and compact matrix quantum groups such as the existence of a Haar state, the representation theory and Woronowicz’s quantum version of the Tannaka–Krein theorem. Building on this, we define Banica–Speicher quantum groups (also called easy quantum groups), a class of compact matrix quantum groups determined by the combinatorics of set partitions. We sketch the classification of Banica–Speicher quantum groups and we list some applications. We review the state-of-the-art regarding Banica–Speicher quantum groups and we list some open problems.

We prove that the L 1-algebra of any non-Kac type compact quantum group is not operator biflat. Since operator amenability implies operator biflatness, this result shows that any co-amenable, non-Kac type compact quantum group gives a counterexample to the conjecture that L 1(𝔾) is operator amenable if and only if 𝔾 is amenable and co-amenable for any locally compact quantum group 𝔾. The result also implies that the L 1-algebra of a locally compact quantum group is operator biprojective if and only if 𝔾 is compact and of Kac type.

Symmetries of Lévy processes on compact quantum groups, their Markov semigroups and potential theory

In this paper, we investigate the co-amenability of compact quantum groups. Combining with some properties of regular C*-norms on algebraic compact quantum groups, we show that the quantum double of co-amenable compact quantum groups is unique. Based on this, this paper proves that co-amenability is preserved under formulation of the quantum double construction of compact quantum groups, which exhibits a type of nice symmetry between the co-amenability of quantum groups and the amenability of groups.

In the theory of representations of a finite group by linear transformations the closely related notions of imprimitivity and of play a prominent role. In [18] the author has generalized these notions to the case in which the group is a separable locally compact topological group and the linear transformations are unitary transformations in Hilbert space. It turns out (and this is the principle theorem of [18]) that the classical theorem of Frobenius according to which every imprimitive representation of a finite group is in a certain canonical fashion by a representation of a subgroup may be reformulated so as to remain true under the more general circumstances indicated above. This connection between representations of groups and representations of their subgroups has many interesting and useful properties in the finite case and it naturally occurs to one to study the extent to which these properties persist in general. The present paper is the first of a projected series in which it is planned to investigate this question in a systematic manner. Our principal results, formulated as Theorems 12.1, 12.2 and 13.1, are closely related; each being essentially a corollary of its predecessor. The first asserts that if L is a representation of the closed subgroup G1 of (M and UL is the corresponding induced representation of (M then the restriction of UL to the closed subgroup G2 is a sum over the G1: G2 double cosets of certain induced representations o G 2 . The second gives a similar decomposition of the Kronecker product UL f? UM where L and M are representations of G1 and G2 respectively. The third provides a usable formula for computing the strong intertwining numbers of the induced representations UL and UM of (M. The in question are ordinary discrete sums only when there are at most countably many non trivial double cosets. In general direct integrals as defined by von Neumann in [26] must be used and we must restrict ourselves to the case in which the relevant double coset decomposition of (M is measurable. As we have shown in detail elsewhere [20] these theorems for finite groups imply certain classical results; in particular the Frobenius reciprocity theorem and the Shoda criteria for the irreducibility and unitary equivalence of monomial representations. Our theorems yield generalizations of these results but these generalizations may be regarded as satisfactory only insofar as they deal with representations whose irreducible constituents are discrete and finite dimensional. More far reaching generalizations must be sought in other directions. We

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a noncommutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SU q(2) arising from measures on its spectrum.

We give an elementary introduction to the theory of algebraic and topological quantum groups (in the spirit of S. L. Woronowicz). In particular, we recall the basic facts from Hopf (*-) algebra theory, theory of compact (matrix) quantum groups and the theory of their actions on compact quantum spaces. We also provide the most important examples, including the classification of quantum SL(2)-groups, their real forms and quantum spheres. We also consider quantum SLq(N)-groups and quantum Lorentz groups.

We compare algebraic objects related to a compact quantum group action on a unital C*-algebra in the sense of Podleś and Baum et al. and show that they differ by the kernel of the morphism describing the action. Then we address ways to remove the kernel without changing the Podleś algebraic core. A minimal such procedure is described. We end the paper with a natural example of an action of a reduced compact quantum group with non-trivial kernel.

For a non-degenerate pair of compact quantum groups, we first construct the quantum double as an algebraic compact quantum group in an algebraic framework. Then by adopting some completion procedure, we give the universal and reduced quantum double constructions in the correspondence C*-algebraic settings, which generalize Drinfeld’s quantum double construction and yield new C*-algebraic compact quantum groups.

Quantum groups have been studied intensively for the last two decades from various points of view. The underlying mathematical structure is that of an algebra with a coproduct. Compact quantum groups admit Haar measures. However, if we want to have a Haar measure also in the noncompact case, we are forced to work with algebras without identity, and the notion of a coproduct has to be adapted. These considerations lead to the theory of multiplier Hopf algebras, which provides the mathematical tool for studying noncompact quantum groups with Haar measures. I will concentrate on the *-algebra case and assume positivity of the invariant integral. Doing so, I create an algebraic framework that serves as a model for the operator algebra approach to quantum groups. Indeed, the theory of locally compact quantum groups can be seen as the topological version of the theory of quantum groups as they are developed here in a purely algebraic context.

For a locally compact quantum group $\mathbb{G}$ we define its center, $\mathscr{Z}(\mathbb{G})$, and its quantum group of inner automorphisms, $\mathrm{Inn}(\mathbb{G})$. We show that one obtains a natural isomorphism between $\mathrm{Inn}(\mathbb{G})$ and $\mathbb{G}/\!\mathscr{Z}(\mathbb{G})$, we characterize normal quantum subgroups of a compact quantum group as those left invariant by the action of the quantum group of inner automorphisms and discuss several examples.