We use a tensor C*-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a *-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal C*-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the C*-algebra in question. The quantum groups involved are A_u(Q) and B_u(Q).
A new approach to the analysis of the physical state space of a theory is presented within the general setting of local quantum physics. It also covers theories with long range forces, such as Quantum Electrodynamics. Making use of the notion of charge class, an extension of the concept of superselection sector, infrared problems are avoided by restricting the states to observables localized in a light cone. The charge structure of a theory can be explored in a systematic manner. The present analysis focuses on simple charges, thus including the electric charge. It is shown that any such charge has a conjugate charge. There is a meaningful concept of statistics: the corresponding charge classes are either of Bose or of Fermi type. The family of simple charge classes is in one--to--one correspondence with the irreducible unitary representations of a compact Abelian group. Moreover, there is a meaningful definition of covariant charge classes. Any such class determines a continuous unitary representation of the Poincare group or its covering group satisfying the relativistic spectrum condition. The resulting particle aspects are also briefly discussed.
Let G be a classical compact Lie group and G_mu the associated compact matrix quantum group deformed by a positive parameter mu (or a nonzero and real mu in the type A case). It is well known that the category Rep(G_mu) of unitary f.d. representations of G_mu is a braided tensor C*-category. We show that any braided tensor *-functor from Rep(G_mu) to another braided tensor C*-category with irreducible tensor unit is full if |mu| eq 1. In particular, the functor of restriction to the representation category of a proper compact quantum subgroup, cannot be made into a braided functor. Our result also shows that the Temperley--Lieb category generated by an object of dimension >2 can not be embedded properly into a larger category with the same objects as a braided tensor C*-subcategory.
To a proper inclusion Nsubset M of II_1 factors of finite Jones index [M:N], we associate an ergodic C*-action of the quantum group S_mu U(2). The deformation parameter is determined by -1<mu<0 and [M:N]=|mu+mu^{-1}|. The higher relative commutants can be identified with the spectral spaces of the tensor powers of the defining representation of the quantum group. This ergodic action may be thought of as a virtual subgroup of S_mu U(2) in the sense of Mackey arising from the tensor category generated by M regarded as a bimodule over N. mu is negative as M is a real bimodule.
This paper addresses the problem of describing the structure of tensor C*-categories M with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on amenability is made. Our assumptions are motivated by the remark that these categories often contain non-full tensor C*-subcategories with conjugates and the same objects admitting an embedding into the Hilbert spaces. Such an embedding defines a compact quantum group by Woronowicz duality. An important example is the Temperley--Lieb category canonically contained in a tensor C*-category generated by a single real or pseudoreal object of dimension bigger than 2. The associated quantum groups are the universal orthogonal quantum groups of Wang and Van Daele. Our main result asserts that there is a full and faithful tensor functor from M to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the G-equivariant Hermitian bundles over compact homogeneous G-spaces, with G a compact group. Our structural results shed light on the problem of whether there is an embedding functor of M into the Hilbert spaces. We show that this is related to the problem of whether a classical compact Lie group can act ergodically on a non-type I von Neumann algebra. In particular, combining this with a result of Wassermann shows that an embedding exists if M is generated by a pseudoreal object of dimension 2.
In algebraic quantum field theory the spacetime manifold is replaced by a suitable base for its topology ordered under inclusion. We explain how certain topological invariants of the manifold can be computed in terms of the base poset. We develop a theory of connections and curvature for bundles over posets in search of a formulation of gauge theories in algebraic quantum field theory.
