The intertwiner spaces of non-easy group-theoretical quantum groups

In 2015, Raum and Weber gave a definition of group-theoretical quantum groups, a class of compact matrix quantum groups with a certain presentation as semi-direct product quantum groups, and studied the case of easy quantum groups. In this article we determine the intertwiner spaces of non-easy group-theoretical quantum groups. We generalise group-theoretical categories of partitions and use a fiber functor to map partitions to linear maps which is slightly different from the one for easy quantum groups. We show that this construction provides the intertwiner spaces of group-theoretical quantum groups in general.Comment: 26 pages; A missing assumption in the statement of Theorem 4.20 has been added. In Remark 4.22 we explain the structure of the intertwiner spaces if this assumption is not satisfied. Several other minor corrections and updates have been mad

Symmetric Hilbert spaces arising from species of structures

Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some `one particle space' \K are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. Any such species $F$ gives rise to an endofunctor \G_F of the category of Hilbert spaces with contractions mapping a Hilbert space \K to a symmetric Hilbert space \G_F(\K) with the same symmetry as the species $F$. A general framework for annihilation and creation operators on these spaces is developed, and compared to the generalised Brownian motions of R. Speicher and M. Bo\.zejko. As a corollary we find that the commutation relation $a_ia_j^*-a_j^*a_i=f(N)\delta_{ij}$ with $Na_i^*-a_i^*N=a_i^*$ admits a realization on a symmetric Hilbert space whenever $f$ has a power series with infinite radius of convergence and positive coefficients.Comment: 39 page

Generalised Brownian Motion and Second Quantisation

A new approach to the generalised Brownian motion introduced by M. Bozejko and R. Speicher is described, based on symmetry rather than deformation. The symmetrisation principle is provided by Joyal's notions of tensorial and combinatorial species. Any such species V gives rise to an endofunctor F_V of the category of Hilbert spaces with contractions. A generalised Brownian motion is an algebra of creation and annihilation operators acting on F_V(H) for arbitrary Hilbert spaces H and having a prescription for the calculation of vacuum expectations in terms of a function t on pair partitions. The positivity is encoded by a *-semigroup of "broken pair partitions" whose representation space with respect to t is V. The existence of the second quantisation as functor Gamma_t from Hilbert spaces to noncommutative probability spaces is proved to be equivalent to the multiplicative property of the function t. For a certain one parameter interpolation between the fermionic and the free Brownian motion it is shown that the ``field algebras'' Gamma(K) are type II_1 factors when K is infinite dimensional.Comment: 33 pages, 5 figure