English fiddling 1650-1850 : reconstructing a lost idiom

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A rigidity result for extensions of braided tensor C*-categories derived from compact matrix quantum groups

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|\neq 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.Comment: 19 pages; published version, to appear in CMP; for a more detailed exposition see v

Regular Objects, Multiplicative Unitaries and Conjugation

The notion of left (resp. right) regular object of a tensor C*-category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (resp. right) regular object, it can be interpreted as a category of corepresentations (resp. representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry. Conjugation is discussed in the context of multiplicative unitaries and their associated Hopf C*-algebras. It is shown that the conjugate of a left regular object is a right regular object in the same category. Furthermore the representation category of a locally compact quantum group has a conjugation. The associated multiplicative unitary is a regular object in that category.Comment: 48 pages, Late

New Light on Infrared Problems: Sectors, Statistics, Symmetries and Spectrum

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 Poincar\'e group or its covering group satisfying the relativistic spectrum condition. The resulting particle aspects are also briefly discussed.Comment: 46 pages, 1 figure, v3 + v4 further references added; version as to appear in Commun. Math. Phy