We introduce the notion of rational Hopf algebras that we think are able to describe the superselection symmetries of two dimensional rational quantum field theories. As an example we show that a six dimensional rational Hopf algebra H can reproduce the fusion rules, the conformal weights, the quantum dimensions and the representation of the modular group of the chiral Ising model. H plays the ...

An unpublished result by the first author states that there exists a Hopf algebra H such that for any Möbius category C (in the sense of Leroux) there exists a canonical algebra morphism from the dual H∗ of H to the incidence algebra of C. Moreover, the Möbius inversion principle in incidence algebras follows from a ‘master’ inversion result in H∗. The underlying module of H was originally defi...

Integrals in Hopf algebras are an essential tool in studying finite dimensional Hopf algebras and their action on rings. Over fields it has been shown by Sweedler that the existence of integrals in a Hopf algebra is equivalent to the Hopf algebra being finite dimensional. In this paper we examine how much of this is true for Hopf algebras over rings. We show that over any commutative ring R tha...

We introduce a quantum double quasitriangular quasi-Hopf algebra D(H) associated to any quasi-Hopf algebra H. The algebra structure is a cocycle double cross product. We use categorical reconstruction methods. As an example, we recover the quasi-Hopf algebra of Dijkgraaf, Pasquier and Roche as the quantum double D(G) associated to a finite group G and group 3-cocycle φ. We also discuss D(Ug) as...

We prove that given a bialgebra surjection π : E → H with nilpotent kernel such that H is a Hopf algebra which is formally smooth as a K-algebra, then π has a section which is a right H-colinear algebra homomorphism. Moreover, if H is also endowed with an ad-invariant integral, then the section can be chosen to be H-bicolinear. We also deal with the dual case. As an application, we prove that e...

Let H be `2 (over C) and let B(H) denote the C*-algebra of all bounded operators on H. Fix an orthonormal basis (en) for H. The atomic masa corresponding to this basis is `∞; equivalently, the algebra of all operators that are diagonalized by the basis (en). The projections in `∞ are exactly the projections onto subspaces spanned by a subset of {en}. That is, P(`∞) ∼= P(N) (here P(A) denotes th...

We extend the definition, from the class of abelian groups to a general locally compact group G, of Feichtinger’s remarkable Segal algebra S0(G). In order to obtain functorial properties for non-abelian groups, in particular a tensor product formula, we endow S0(G) with an operator space structure. With this structure S0(G) is simultaneously an operator Segal algebra of the Fourier algebra A(G)...

We consider the structure of JordanH-pseudoalgebras which are linearly finitely generated over a Hopf algebra H . There are two cases under consideration: H = U(h) and H = U(h)#C[Γ], where h is a finite-dimensional Lie algebra over C, Γ is an arbitrary group acting on U(h) by automorphisms. We construct an analogue of the Tits–Kantor–Koecher construction for finite Jordan pseudoalgebras and des...

We show that the Casimir operators of the semidirect products G2 ⊕2Γ(a,b)⊕Γ(0,0)h of the exceptional Lie algebra G2 and a Heisenberg algebra h can be constructed explicitly from the Casimir operators of G2.

We extend Barr’s well-known characterization of the final coalgebra of a Set-endofunctor H as the completion of its initial algebra to the EilenbergMoore category Alg(M) of algebras associated to a Set-monad M, if H can be lifted to Alg(M). As further analysis, we introduce the notion of commuting pair of endofunctors (T,H) with respect to a monad M and show that under reasonable assumptions, t...