33 research outputs found
A remark on the invariance of dimension
Combining Kulpa's proof of the cubical Sperner lemma and a dimension
theoretic idea of van Mill we give a very short proof of the invariance of
dimension, i.e. the statement that cubes [0,1]^n, [0,1]^m are homeomorphic if
and only if n=m. This note is adapted from lecture notes for a course on
general topology.Comment: latex2e, 8 pages, 1 eps figur
On Charged Fields with Group Symmetry and Degeneracies of Verlinde's Matrix S
We consider the complete normal field net with compact symmetry group
constructed by Doplicher and Roberts starting from a net of local observables
in >=2+1 spacetime dimensions and its set of localized (DHR) representations.
We prove that the field net does not possess nontrivial DHR sectors, provided
the observables have only finitely many sectors. Whereas the superselection
structure in 1+1 dimensions typically does not arise from a group, the DR
construction is applicable to `degenerate sectors', the existence of which (in
the rational case) is equivalent to non-invertibility of Verlinde's S-matrix.
We prove Rehren's conjecture that the enlarged theory is non-degenerate, which
implies that every degenerate theory is an `orbifold' theory. Thus, the
symmetry of a generic model `factorizes' into a group part and a pure quantum
part which still must be clarified.Comment: latex2e, 24 pages. Final version, to appear in Ann. Inst. H. Poinc.
(Theor. Phys.). A serious gap in the proof of Prop. 2.3 has been filled in,
but only under a rationality assumption. The main application to rational
CQFTs is not affected, in fact strenghtened by the new Props. 3.8 and 3.14.
Some remarks have been adde
From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence of tensor categories
We consider certain categorical structures that are implicit in subfactor
theory. Making the connection between subfactor theory (at finite index) and
category theory explicit sheds light on both subjects. Furthermore, it allows
various generalizations of these structures, e.g. to arbitrary ground fields,
and the proof of new results about topological invariants in three dimensions.
The central notion is that of a Frobenius algebra in a tensor category A, which
reduces to the classical notion if A=F-Vect, where F is a field. An object X in
A with two-sided dual X^ gives rise to a Frobenius algebra in A, and under weak
additional conditions we prove a converse: There exists a bicategory E with
Obj(E)={X,Y} such that End_E(X,X) is equivalent to A and such that there are J:
Y->X, J^: X->Y producing the given Frobenius algebra. Many properties
(additivity, sphericity, semisimplicity,...) of A carry over to E. We define
weak monoidal Morita equivalence (wMe) of tensor categories and establish a
correspondence between Frobenius algebras in A and tensor categories B wMe A.
While considerably weaker than equivalence of tensor categories, weak monoidal
Morita equivalence of A and B implies (for A,B semisimple and spherical or
*-categories) that A and B have the same dimension, braided equivalent `center'
(quantum double) and define the same state sum invariants of closed oriented
3-manifolds as defined by Barrett and Westbury. If H is a finite dimensional
semisimple and cosemisimple Hopf algebra then H-mod and H^-mod are wMe. The
present formalism permits a fairly complete analysis of the quantum double of a
semisimple spherical category, which is the subject of the companion paper
math.CT/0111205.Comment: latex2e, ca. 58 pages. Requires diagrams.tex V3.88. Proof of Thm.
6.20 and one reference include
Galois Theory for Braided Tensor Categories and the Modular Closure
Given a braided tensor *-category C with conjugate (dual) objects and
irreducible unit together with a full symmetric subcategory S we define a
crossed product C\rtimes S. This construction yields a tensor *-category with
conjugates and an irreducible unit. (A *-category is a category enriched over
Vect_C with positive *-operation.) A Galois correspondence is established
between intermediate categories sitting between C and C\rtimes S and closed
subgroups of the Galois group Gal(C\rtimes S/C)=Aut_C(C\rtimes S) of C, the
latter being isomorphic to the compact group associated to S by the duality
theorem of Doplicher and Roberts. Denoting by D\subset C the full subcategory
of degenerate objects, i.e. objects which have trivial monodromy with all
objects of C, the braiding of C extends to a braiding of C\rtimes S iff
S\subset D. Under this condition C\rtimes S has no degenerate objects iff S=D.
If the original category C is rational (i.e. has only finitely many equivalence
classes of irreducible objects) then the same holds for the new one. The
category C\rtimes D is called the modular closure of C since in the rational
case it is modular, i.e. gives rise to a unitary representation of the modular
group SL(2,Z). (In passing we prove that every braided tensor *-category with
conjugates automatically is a ribbon category, i.e. has a twist.) If all simple
objects of S have dimension one the structure of the category C\rtimes S can be
clarified quite explicitly in terms of group cohomology.Comment: latex2e, 39 pages. Style file included via filecontents command.
Final version, to appear in Adv. Math. Purely notational improvement
Tensor categories: A selective guided tour
These are the lecture notes for a short course on tensor categories. The
coverage in these notes is relatively non-technical, focussing on the essential
ideas. They are meant to be accessible for beginners, but it is hoped that also
some of the experts will find something interesting in them.
Once the basic definitions are given, the focus is mainly on k-linear
categories with finite dimensional hom-spaces. Connections with quantum groups
and low dimensional topology are pointed out, but these notes have no
pretension to cover the latter subjects at any depth. Essentially, these notes
should be considered as annotations to the extensive bibliography.Comment: 57 pages. Notes for a three-hour lecture course. Extensively revised,
updated and extended. To appear in Revista de la Uni\'on Matem\'atica
Argentin