Residually finite quantum group algebras

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsion. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C ∗ \mathrm {C}^{*} -algebra of a countable discrete quantum group \mathbbl {\Gamma } implies that \mathbbl {\Gamma } is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition.

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.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

In this paper, the complete algebraic structure of the finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.

The wedge-direct sum of a sequence of semisimple algebras (over an arbitrary field) was introduced in [Xu, B. (2019). Wedge-direct sums of table algebras and applications to association schemes, I. Commun. Algebra 47(8):3309–3328]. The group algebras (over the complex field) of finite groups are a special class of semisimple algebras. In [Xu, B. (2020). Wedge-direct sums of table algebras and applications to association schemes, II. J. Pure Appl. Algebra 224(11):106402], the wedge-direct sum of a sequence of group algebras of finite p-groups is used to obtain the characterization and classification of a class of p-table algebras. In this article, we continue the research in above mentioned articles and characterize, in terms of table algebras, the wedge-direct sum of a sequence of group algebras of arbitrary finite groups. Applications to association schemes will also be discussed.

Supports of irreducible spherical representations of rational Cherednik algebras of finite Coxeter groups

Idempotent states on a compact quantum group are shown to yield group-like projections in the multiplier algebra of the dual discrete quantum group. This allows to deduce that every idempotent state on a finite quantum group arises in a canonical way as the Haar state on a finite quantum hypergroup. A natural order structure on the set of idempotent states is also studied and some examples discussed.

This is a survey of some aspects of the subject of approximation properties for locally compact quantum groups, based on lectures given at the {\it Topological Quantum Groups} Graduate School, 28 June - 11 July, 2015 in Bedlewo, Poland. We begin with a study of the dual notions of amenability and co-amenability, and then consider weakenings of these properties in the form of the Haagerup property and weak amenability. For discrete quantum groups, the interaction between these properties and various operator algebra approximation properties are investigated. We also study the connection between central approximation properties for discrete quantum groups and monoidal equivalence for their compact duals. We finish by discussing the central weak amenability and central Haagerup property for free quantum groups.

Tensorial sets adapted to sequences of finite subgroups are applied to the crystal field problem, and a general method for generating sequence‐adapted molecular tensors using finite group algebra is formulated. All subgroup sequences of the abstract finite group G(24), isomorphic to the octahedral, O, tetrahedral, Td, and symmetric, S(4), groups are tabulated with explicit isomorphisms provided. The sequences fall into eight equivalence classes. A catalog of irreducible representations of G(24) adapted to a member of each of the eight sequence classes is given together with the transformations which generate representations adapted to all other sequences. With this data it is possible to systematically generate tensorial sets adapted to any sequence of a realization of G(24). Unitary transformations which adapt conventional forms of first‐ and second‐rank irreducible tensorial sets of the rotation group to the eight sequences of the octahedral group are provided. Forms suitable for use with magnetic fields are included. The problem of a d1 ion in a trigonal crystal field is treated with sequence‐adapted molecular tensors, and the utility of different sequences for descent in symmetry is discussed.

Quantum automorphism groups of finite quantum groups are classical

Let be either a simple linear algebraic group over an algebraically closed field of characteristic or a quantum group at an ‐th root of unity. We define a tensor ideal of singular ‐ modules in the category of finite‐dimensional ‐modules and study the associated quotient category , called the regular quotient . For , the Coxeter number of , we establish a ‘linkage principle’ and a ‘translation principle’ for tensor products: Let be the essential image in of the principal block of . We first show that is closed under tensor products in . Then we prove that the monoidal structure of is governed to a large extent by the monoidal structure of . These results can be combined to give an external tensor product decomposition , where denotes the Verlinde category of .

Graphs of quantum groups and K-amenability

Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group and 1-injectivity of as an operator -module. In particular, a locally compact group G is amenable if and only if its group von Neumann algebra VN(G) is 1-injective as an operator module over the Fourier algebra A(G). As an application, we provide a decomposability result for completely bounded -module maps on , and give a simpliûed proof that amenable discrete quantum groups have co-amenable compact duals, which avoids the use of modular theory and the Powers-Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.

Nontriviality of the first Hochschild cohomology of some block algebras of finite groups

I construct an algebraic model for a typical fiber on a 1 + 1 dimensional spacetime. The vector space comprising the fiber is composed of elements x ⊗ x formed from the direct product of two copies of an element x in the D 2 = C 2 ⊗ C 2 finite group algebra over the real numbers. The fiber contains subspaces whose elements can be associated with the tangent and momentum vectors of trajectories in the manifold. The fiber also contains a subspace whose elements are associated with the local flow of action of each trajectory. The condition of minimum action translates into a constraint on the original vector x in the direct product structure.KeywordsTangent spacemanifoldfiber differential geometry1 + 1 spacetime