Florian Nill

DIAGONAL CROSSED PRODUCTS BY DUALS OF QUASI-QUANTUM GROUPS

A two-sided coaction of a Hopf algebra on an associative algebra ℳ is an algebra map of the form , where (λ,ρ) is a commuting pair of left and right -coactions on ℳ, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra on ℳ by ◃ and ▹, respectively, we define the diagonal crossed product to be the algebra generated by ℳ and with relations given by We give a natural generalization of this construction to the case where is a quasi-Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e. where the coproduct Δ is non-unital). In these cases our diagonal crossed product will still be an associative algebra structure on extending , even though the analogue of an ordinary crossed product in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasi-quantum group symmetry given by G. Mack and V. Schomerus [31, 47] as well as the formulation of Hopf spin chains or lattice current algebras based on truncated quantum groups at roots of unity. In the case and λ=ρ=Δ we obtain an explicit definition of the quantum double for quasi-Hopf algebras , which before had been described in the form of an implicit Tannaka–Krein reconstruction procedure by S. Majid [35]. We prove that is itself a (weak) quasi-bialgebra and that any diagonal crossed product naturally admits a two-sided -coaction. In particular, the above-mentioned lattice models always admit the quantum double as a localized cosymmetry, generalizing results of Nill and Szlachanyi [42]. A complete proof that is even a (weak) quasi-triangular quasi-Hopf algebra will be given in a separate paper [27].

QUANTUM CHAINS OF HOPF ALGEBRAS WITH QUANTUM DOUBLE COSYMMETRY

Abstract:Given a finite dimensional C*-Hopf algebra H and its dual Ĥ we construct the infinite crossed product and study its superselection sectors in the framework of algebraic quantum field theory. is the observable algebra of a generalized quantum spin chain with H-order and Ĥ-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If is a group algebra then becomes an ordinary G-spin model. We classify all DHR-sectors of – relative to some Haag dual vacuum representation – and prove that their symmetry is described by the Drinfeld double . To achieve this we construct localized coactions and use a certain compressibility property to prove that they are universal amplimorphisms on . In this way the double can be recovered from the observable algebra as a universal cosymmetry.

Fusion structures from quantum groups. II: Why truncation is necessary

It is shown that a finite, reflection positive, and nontruncated fusion structure on an arbitrary Hopf algebraℋ is trivial in the sense thatq-traces coincide with ordinary traces andq-dimensions coincide with ordinary dimensions. Thus, nontruncated fusion structures are ruled out to describe the fusion rules of quantum field theories with noninteger statistical dimensions and a finite number of superselection sectors.

A Comment on Jones Inclusions with infinite Index

Given an irreducible inclusion of infinite von-Neumann-algebras N ⊂ M together with a conditional expectation E : M → M such that the inclusion has depth 2, we show quite explicitely how N can be viewed as the fixed point algebra of M w.r.t. an outer action of a compact Kac-algebra acting on M. This gives an alternative proof, under this special setting, of a more general result of M. Enock and R. Nest, [E-N], see also S. Yamagami, [Ya2].

Dyonic Sectors and Intertwiner Connections in 2+1-Dimensional Lattice \(\)-Higgs Models

Abstract:We construct dyonic states in 2+1-dimensional lattice -Higgs models, i.e. states which are both, electrically and magnetically charged. These states are parametrized by , where ɛ and μ are -valued electric and magnetic charge distributions, respectively, living on the spatial lattice . The associated Hilbert spaces carry charged representations of the observable algebra , the global transfer matrix t and a unitary implementation of the group of spatial lattice translations. We prove that for coinciding total charges these representations are dynamically equivalent and we construct a local intertwiner connection , where is a path in the space of charge distributions . The holonomy of this connection is given by -valued phases. This will be the starting point for a construction of scattering states with anyon statistics in a subsequent paper.

The Coleman-Weinberg Mechanism in the Abelian Lattice Higgs Model

The order of the Higgs phase transition in the Abelian lattice Higgs model is investigated by means of perturbation theory. Based on gauge invariant criteria such as generalized Clausius-Clapeyron and Ehrenfest equations it is shown that in the validity domain of the loop expansion, i.e., e → 0 and Ne2 fixed, there is no evidence for a first order transition up to two loops. Here e is the gauge coupling and λ the scalar self coupling constant. If, however, we perform a perturbation analysis as e → 0 and Ne4 is kept fixed, the Coleman-Weinberg-Linde result on a first order phase transition can be rederived gauge invariantly. This leads to an intimate relation between Linde’s lower bound on the Higgs mass in a continuum theory and the possible existence of a tricritical line separating the first order from the second order phase transition in the lattice model. In tems of renormalized parameters the tricritical line is given explicitly up to second order. I close with a tentative discussion of different scenarios as to where the tricritical line may be found in the space of bare coupling constants.