Axioms for Weak Bialgebras

Let A be a finite dimensional unital associative algebra over a field K, which is also equipped with a coassociative counital coalgebra structure (\Delta,\eps). A is called a Weak Bialgebra if the coproduct \Delta is multiplicative. We do not require \Delta(1) = 1 \otimes 1 nor multiplicativity of the counit \eps. Instead, we propose a new set of counit axioms, which are modelled so as to guarantee that \Rep\A becomes a monoidal category with unit object given by the cyclic A-submodule \E := (A --> \eps) \subset \hat A (\hat A denoting the dual weak bialgebra). Under these monoidality axioms \E and \bar\E := (\eps <-- A) become commuting unital subalgebras of \hat A which are trivial if and only if the counit \eps is multiplicative. We also propose axioms for an antipode S such that the category \Rep\A becomes rigid. S is uniquely determined, provided it exists. If a monoidal weak bialgebra A has an antipode S, then its dual \hat A is monoidal if and only if S is a bialgebra anti-homomorphism, in which case S is also invertible. In this way we obtain a definition of weak Hopf algebras which in Appendix A will be shown to be equivalent to the one given independently by G. B\"ohm and K. Szlach\'anyi. Special examples are given by the face algebras of T. Hayashi and the generalised Kac algebras of T. Yamanouchi.Comment: 48 pages, Late

On the Structure of Monodromy Algebras and Drinfeld Doubles

We give a review and some new relations on the structure of the monodromy algebra (also called loop algebra) associated with a quasitriangular Hopf algebra H. It is shown that as an algebra it coincides with the so-called braided group constructed by S. Majid on the dual of H. Gauge transformations act on monodromy algebras via the coadjoint action. Applying a result of Majid, the resulting crossed product is isomorphic to the Drinfeld double D(H). Hence, under the so-called factorizability condition given by N. Reshetikhin and M. Semenov-Tian- Shansky, both algebras are isomorphic to the algbraic tensor product H\otimes H. It is indicated that in this way the results of Alekseev et al. on lattice current algebras are consistent with the theory of more general Hopf spin chains given by K. Szlach\'anyi and the author. In the Appendix the multi-loop algebras L_m of Alekseev and Schomerus [AS] are identified with braided tensor products of monodromy algebras in the sense of Majid, which leads to an explanation of the ``bosonization formula'' of [AS] representing L_m as H\otimes\dots\otimes H.Comment: Latex, 22 p., revised Oct.6, 1996, some references added, more historical background in the introduction, some minor technical improvements, E-mail: [email protected]

Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Given a finite dimensional C^*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H ... and study its superselection sectors in the framework of algebraic quantum field theory. A is the observable algebra of a generalized quantum spin chain with H-order and H^-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If H=\CC G is a group algebra then A becomes an ordinary G-spin model. We classify all DHR-sectors of A --- relative to some Haag dual vacuum representation --- and prove that their symmetry is described by the Drinfeld double D(H). To achieve this we construct localized coactions \rho: A \to (A \otimes D(H)) and use a certain compressibility property to prove that they are universal amplimorphisms on A. In this way the double D(H) can be recovered from the observable algebra A as a universal cosymmetrty.Comment: Latex, 48 pages, no figures, extended version of hep-th/9507174, but without the field algebra construction, contains full proofs of the slightly shortened article published in Commun.Math.Phys., the revision only concerns some misprints and an update of the literatur

A Comment on Jones Inclusions with infinite Index

Given an irreducible inclusion of infinite von-Neumann-algebras \cn \subset \cm together with a conditional expectation E : \cm \rightarrow \cm such that the inclusion has depth 2, we show quite explicitely how \cn can be viewed as the fixed point algebra of \cm w.r.t. an outer action of a compact Kac-algebra acting on \cm. 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].Comment: latex, 40 page

On the Redundancy of Birth and Death Rates in Homogeneous Epidemic SIR Models

The dynamics of fractional population sizes =/ in homogeneous compartment models with time-dependent total population N is analyzed. Assuming constant per capita birth and death rates, the vector field ˙=() naturally projects to a vector field () tangent to the leaves of constant population N. A universal formula for the projected field is given. In this way, in many SIR-type models with standard incidence, all demographic parameters become redundant for the dynamical system ˙=(). They may be put to zero by shifting the remaining parameters appropriately. Normalizing eight examples from the literature this way, they unexpectedly become isomorphic for corresponding parameter ranges. Thus, some recently published results turn out to have been covered already by papers 20 years ago

On the redundancy of birth and death rates in homogenous epidemic SIR models

The dynamics of fractional population sizes Y_i/N in homogeneous compartment models is analyzed. Assuming constant per capita birth and death rates a universal formula for the projected vector field tangent to the leaves of constant population N is given. In this way, in many SIR-type models with standard incidence all demographic parameters become redundant and may be put to zero by shifting remaining parameters. Normalizing eight examples from the literature this way, they unexpectedly become isomorphic for corresponding parameter ranges. Thus, some recently published results turn out to be already covered by papers 20 years ago.Comment: 6 pages, 1 figure, 2 Table

Scaling Symmetries and Parameter Reduction in Epidemic SI(R)S Models

Symmetry concepts in parametrized dynamical systems may reduce the number of external parameters by a suitable normalization prescription. If, under the action of a symmetry group G , parameter space A becomes a (locally) trivial principal bundle, A ≅ A / G × G , then the normalized dynamics only depends on the quotient A / G . In this way, the dynamics of fractional variables in homogeneous epidemic SI(R)S models, with standard incidence, absence of R-susceptibility and compartment independent birth and death rates, turns out to be isomorphic to (a marginally extended version of) Hethcote’s classic endemic model, first presented in 1973. The paper studies a 10-parameter master model with constant and I-linear vaccination rates, vertical transmission and a vaccination rate for susceptible newborns. As recently shown by the author, all demographic parameters are redundant. After adjusting time scale, the remaining 5-parameter model admits a 3-dimensional abelian scaling symmetry. By normalization we end up with Hethcote’s extended 2-parameter model. Thus, in view of symmetry concepts, reproving theorems on endemic bifurcation and stability in such models becomes needless