Narrow Results

We prove Haag duality for cone-like regions in the ground state representation corresponding to the translational invariant ground state of Kitaev’s quantum double model for finite abelian groups. This property says that if an observable commutes with all observables localized outside the cone region, it actually is an element of the von Neumann algebra generated by the local observables inside the cone. This strengthens locality, which says that observables localized in disjoint regions commute.

As an application, we consider the superselection structure of the quantum double model for abelian groups on an infinite lattice in the spirit of the Doplicher–Haag–Roberts program in algebraic quantum field theory. We find that, as is the case for the toric code model on an infinite lattice, the superselection structure is given by the category of irreducible representations of the quantum double.

We introduce fusion, contragradient and braiding of Hilbert affine representations of a subfactor planar algebra P (not necessarily having finite depth). We prove that if N ⊂ M is a subfactor realization of P, then the Drinfeld center of the N–N-bimodule category generated by _NL²(M)_M, is equivalent to the category of Hilbert affine representations of P satisfying certain finiteness criterion. As a consequence, we prove Kevin Walker's conjecture for planar algebras.

Let A be a finite-dimensional oriented quantum algebra over a field k There an oriented quantum algebra structure on A⊗A which is motivated by an algebra isomorphism of the quantum double D(A) with A⊗A when A has a factorizable Hopf algebra structure.

We describe and study the oriented quantum algebra structure on A⊗A and work it out in detail when A = M₂(k) is the algebra of 2×2 matrices over k and has the oriented quantum algebra structure which gives rise to the Jones polynomial. The resulting 16×16 braiding matrix gives rise to a non-trivial ambient isotopy invariant of oriented links.

Suppose that A is a finite-dimensional factorizable Hopf algebra over k and that the double D(A) has a ribbon element. Then the Hennings invariant of 3-manifolds is defined for D(A) and can be computed in a very natural way terms of an oriented quantum algebra structure on A⊗A. We supply details for future calculations which are developed in [2].

We obtain an R-matrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)-Lie algebra or braided-Lie algebra. The same result applies for every (super)-Hopf algebra or braided-Hopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a non-trivial one on the ring k[x] of polynomials in one variable, regarded as a braided-line. Representations of the extended Artin braid group for braids in the complement of S¹ are also obtained by the same method.

Let k be an algebraically closed field of odd characteristic p, and let D_n be the dihedral group of order 2n such that p|2n. Let D(kD_n) denote the quantum double of the group algebra kD_n. In this paper, we describe the structures of all finite-dimensional indecomposable left D(kD_n)-modules, equivalently, of all finite-dimensional indecomposable Yetter-Drinfeld kD_n-modules, and classify them.

The theory of weak Hopf algebras is a development of the theory of Hopf algebras, which is important in the theory of quantum groups and the related theory of mathematical physics. In this paper, we give a survey of this theory and its applications.

Firstly, we review some concepts related to weak Hopf algebras and their relationship and moreover, some algebraical structures. Secondly, we collect the generalizations of quantum double, i.e. quantum quasi-double and quantum G-double, constructed from some certain weak Hopf algebras so as to obtain a class of singular solutions of the quantum Yang-Baxter equation (QYBE). Their properties are also introduced in this part. Thirdly, as the important examples of weak Hopf algebras, some weaken quantized enveloping algebra are given which are associated with some generalizations of Lie algebra, such as generalized Kac-Moody algebra, Borcherds superalgebras. Moreover, their algebraic structures are discussed such that one can understand them clearly. Finally, some generalized notions of tensor category are introduced including pre-tensor category and weak tensor category and their braidings. They can be realized respectively from the categories of representations of the related classes of weak Hopf algebras and their quantum quasi-doubles. As the categorical version of a (singular) solutions of the QYBE, a general Yang-Baxter operator is studied. In summary, we list the diagrams of relations among these kinds of categories.