On Positive Definiteness Over Locally 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.

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 consider approximation methods defined by translates of a positive definite function on a compact group. A characterization of the native space generated by a positive definite function on a compact group is presented. Starting from Bochner's theorem, we construct examples of well-localized positive definite central functions on the rotation group (3). Finally, the stability of the interpolation problem and the error analysis for the given examples are studied in detail.

Examples of non-compact quantum group actions

On square roots of the Haar state on compact quantum groups

We consider a coamenable compact quantum group \mathbb{G} as a compact quantum metric space if its function algebra \mathrm{C}(\mathbb{G}) is equipped with a Lip-norm. By using a projection P onto direct summands of the Peter–Weyl decomposition, the \mathrm{C}^{*} -algebra \mathrm{C}(\mathbb{G}) can be compressed to an operator system P\mathrm{C}(\mathbb{G})P , and there are induced left and right coactions on this operator system. Assuming that the Lip-norm on \mathrm{C}(\mathbb{G}) is bi-invariant in the sense of Li, there is an induced bi-invariant Lip-norm on the operator system P\mathrm{C}(\mathbb{G})P turning it into a compact quantum metric space. Given an appropriate net of such projections which converges strongly to the identity map on the Hilbert space \mathrm{L}^{2}(\mathbb{G}) , we obtain a net of compact quantum metric spaces. We prove convergence of such nets in terms of Kerr’s complete Gromov–Hausdorff distance. An important tool is the choice of an appropriate state whose induced slice map gives an approximate inverse of the compression map \mathrm{C}(\mathbb{G}) \ni a \mapsto PaP in Lip-norm.

Convolution semigroups of states on a quantum group form the natural noncommutative analogue of convolution semigroups of probability measures on a locally compact group. Here we initiate a theory of weakly continuous convolution semigroups of functionals on a C*-bialgebra, the noncommutative counterpart of a locally compact semigroup. On locally compact quantum groups we obtain a bijective correspondence between such convolution semigroups and a class of C 0-semigroups of maps which we characterise. On C*-bialgebras of discrete type we show that all weakly continuous convolution semigroups of states are automatically norm-continuous. As an application we deduce a known characterisation of continuous conditionally positive-definite Hermitian functions on a compact group.

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.

Stability results for approximation by positive definite functions on SO(3)

The canonical mapping on the product of a LCA group with its dual is shown to extend uniquely in a homomorphic and analytic way to the product of the respective complexifications.

Non-existence of genuine (compact) quantum symmetries of compact, connected smooth manifolds

Embeddable quantum homogeneous spaces

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.

Fourier–Stieltjes Algebras of Locally Compact Groupoids

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