Quantum automorphism groups of finite quantum groups are classical

On square roots of the Haar state on compact quantum groups

In this paper we study the states of Poisson type and infinitely divisible states on compact quantum groups. Each state of Poisson type is infinitely divisible, i.e., it admits n-th root for all nge 1. The main result is that on finite quantum groups infinitely divisible states must be of Poisson type. This generalizes Böge’s theorem concerning infinitely divisible measures (commutative case) and Parthasarathy’s result on infinitely divisible positive definite functions (cocommutative case). Two proofs are given.

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.

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.

Quantum and braided diffeomorphism groups

Answering a question of Shuzhou Wang we give a description of quantum \mathrm{SO}(3) groups of Podleś as universal compact quantum groups acting on the \mathrm{C}^* -algebra M_2 and preserving the Powers state. We use this result to give a complete classification of all continuous compact quantum group actions on M_2 .

Verma and simple modules for quantum groups at non-abelian groups

We introduce a notion of partition wreath product of a finite group by a partition quantum group, a construction motivated on the one hand by classical wreath products and on the other hand by the free wreath product of J. Bichon.We identify the resulting quantum group in several cases, establish some of its properties and show that when the finite group in question is abelian, the partition wreath product is itself a partition quantum group. This allows us to compute its representation theory, using earlier results of the first named author.

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.

Sekine quantum groups are a family of finite quantum groups. The main result of this article is to compute all the idempotent states on Sekine quantum groups, which completes the work of Franz and Skalski. This is achieved by solving a complicated system of equations using linear algebra and basic number theory. From this, we discover a new class of non-Haar idempotent states. The order structure of the idempotent states on Sekine quantum groups is also discussed. Finally we give a sufficient condition for the convolution powers of states on Sekine quantum group to converge.

Residually finite quantum group algebras

We find a finite free resolution of the counit of the free unitary quantum groups of van Daele and Wang and, more generally, Bichon's universal cosovereign Hopf algebras with a generic parameter matrix. This allows us to compute Hochschild cohomology with one‐dimensional coefficients for all these Hopf algebras. In fact, the resolutions can be endowed with a Yetter–Drinfeld structure. General results of Bichon then allow us to compute also the corresponding bialgebra cohomologies. Finding the resolution rests on two pillars. We take as a starting point the resolution for the free orthogonal quantum group presented by Collins, Härtel, and Thom or its algebraic generalization to quantum symmetry groups of bilinear forms due to Bichon. Then, we make use of the fact that the free unitary quantum groups and some of its non‐Kac versions can be realized as a glued free product of a (non‐Kac) free orthogonal quantum group with , the finite group of order 2. To obtain the resolution also for more general universal cosovereign Hopf algebras, we extend Gromada's proof from compact quantum groups to the framework of matrix Hopf algebras. As a by‐product of this approach, we also obtain a projective resolution for the freely modified bistochastic quantum groups. Only a special subclass of free unitary quantum groups and universal cosovereign Hopf algebras decompose as a glued free product in the described way. In order to verify that the sequence we found is a free resolution in general (as long as the parameter matrix is generic, two conditions which are automatically fulfilled in the free unitary quantum group case), we use the theory of Hopf bi‐Galois objects and Bichon's results on monoidal equivalences between the categories of Yetter–Drinfeld modules over universal cosovereign Hopf algebras for different parameter matrices.

The existence of Miura-type free field realizations is established for the extended conformal algebrasW(sl(n)) at irrational values of the screening parameter. The problem of the “closure” of the algebra is reduced to a finite dimensional quantum group problem. The structure of the Fock space resolution and the character formula are obtained for the irreducible modules. As graded vector spaces these modules are shown to be isomorphic to the space ofsl(n) singlets in\(\widehat{sl}(n)\) affine level 1 modules. The isomorphism is given by the ϕβγ free field realization of\(\widehat{sl}(n)\).

In [3] we initiated our study of the automorphism groups of a certain class of near-rings. Specifically, let P be any complex polynomial and let P denote the near-ring of all continuous selfmaps of the complex plane where addition of functions is pointwise and the product fg of two functions f and g in P is defined by fg=f∘P∘g. The near-ring P is referred to as a laminated near-ring with laminating element P. In [3], we characterised those polynomials P(z)=anzn + an−1zn−1 +…+a0 for which Aut P is a finite group. We are able to show that Aut P is finite if and only if Deg P≧3 and ai ≠ 0 for some i ≠ 0, n. In addition, we were able to completely determine those infinite groups which occur as automorphism groups of the near-rings P. There are exactly three of them. One is GL(2) the full linear group of all real 2×2 nonsingular matrices and the other two are subgroups of GL(2). In this paper, we begin our study of the finite automorphism groups of the near-rings P. We get a result which, in contrast to the situation for the infinite automorphism groups, shows that infinitely many finite groups occur as automorphism groups of the near-rings under consideration. In addition to this and other results, we completely determine Aut P when the coefficients of P are real and Deg P = 3 or 4.

We show that under certain conditions a finite dimensional graded pointed Hopf algebra is an image of an algebra twist of a quantized enveloping algebra Uq(b) when q is aro ot of unity. In addition we obtain a classification of Hopf algebras H such that G(H) ha s odd prime order p > 7 a nd grH is of Cartan type.