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 ∈ 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 ObjE = {A, B} such that EndE(A) ⊗ ≃ A and such that there are J, J: B ⇋ A producing the given Frobenius algebra. Many properties (additivity, sphericity, semisimplicity,...) of A carry over to the bicategory E. We define weak monoidal Morita equivalence of tensor categories, denoted A ≈ B, and establish a correspondence between Frobenius algebras in A and tensor categories B ≈ A. While considerably weaker than equivalence of tensor categories, weak monoidal Morita equivalence A ≈ B has remarkable consequences: A and B have equivalent (as braided tensor categories) quantum doubles (‘centers’) and (if A, B are semisimple spherical or ∗-categories) have equal dimensions and give rise the same state sum invariant of closed oriented 3-manifolds as recently defined by Barrett and Westbury. An instructive example is provided by finite dimensional semisimple and cosemisimple Hopf algebras, for which we prove H − mod ≈ ˆH − mod. The present formalism permits a fairly complete analysis of the center of a semisimple spherical category, which is the subject of the companion paper math.CT/0111205. 1

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Abstract 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 φ (G) associated to a finite group G and group 3-cocycle φ. We also discuss D φ (Ug) associate to a Lie algebra g and Drinfeld’s cocycle φ obtained from a solution of the KZ equation.

In this paper we present explicit formulas for the ∗-product on quantum spaces which are of particular importance in physics, i.e., the q-deformed Minkowski space and the q-deformed Euclidean space in 3 and 4 dimensions, respectively. Our formulas are complete and formulated using the deformation parameter q. In addition, we worked out an expansion in powers of h = ln q up to second order, for all considered cases.

In this article we consider quantum spaces which could be of particular importance in physics, i.e. the 2-dimensional quantum plane, the q-deformed Euclidean space with 3 or 4 dimensions as well as the q-deformed Minkowski space. For each of these spaces we present some standard techniques for dealing with q-deformed Grassmann variables. Especially, we give formulae for multiplying two supernumbers and show how symmetry generators and fermionic derivatives act on antisymmetrized quantum spaces. Furthermore, we review for all types of quantum spaces their Hopf structures. From the corresponding formulae for the coproduct we are then able to read off a realization of the L-matrices in terms of the symmetry generators. This means that the commutation relations between all types of quantum spaces are calculable as soon as the actions of the symmetry generators are known. 1

Abstract Quantum groups emerged in the latter quarter of the 20th century as, on the one hand, a deep and natural generalisation of symmetry groups for certain integrable systems, and on the other as part of a generalisation of geometry itself powerful enough to make sense in the quantum domain. Just as the last century saw the birth of classical geometry, so the present century sees at its end the birth of this quantum or noncommutative geometry, both as an elegant mathematical reality and in the form of the first theoretical predictions for Planck-scale physics via ongoing astronomical measurements. Noncommutativity of spacetime, in particular, amounts to a postulated new force or physical effect called cogravity.

Abstract. We study a twisted version of the Yang-Baxter Equation, called the Hom-Yang-Baxter Equation (HYBE), which is motivated by Hom-Lie algebras. Three classes of solutions of the HYBE are constructed, one from Hom-Lie algebras and the others from Drinfeld’s (dual) quasitriangular bialgebras. Each solution of the HYBE can be extended to operators that satisfy the braid relations. Assuming an invertibility condition, these operators give a representation of the braid group. 1.

Abstract. Motivated by recent work on Hom-Lie algebras, a twisted version of the Yang-Baxter equation, called the Hom-Yang-Baxter equation (HYBE), was introduced by the author in [62]. In this paper, several more classes of solutions of the HYBE are constructed. Some of these solutions of the HYBE are closely related to the quantum enveloping algebra of sl(2), the Jones-Conway polynomial, and Yetter-Drinfel’d modules. We also construct a new infinite sequence of solutions of the HYBE from a given one. Along the way, we compute all the Lie algebra endomorphisms on the (1 + 1)-Poincaré algebra and sl(2). 1.

H, and show that the category of Yetter-Drinfeld modules is isomorphic to

This is an introduction for nonspecialists to the noncommutative geometric approach to Planck scale physics coming out of quantum groups. The canonical role of the ‘Planck scale quantum group ’ C[x]◮⊳C[p] and its observable-state T-dualitylike properties are explained. The general meaning of noncommutativity of position space as potentially a new force in Nature is explained as equivalent under quantum group Fourier transform to curvature in momentum space. More general quantum groups C(G ⋆)◮⊳U(g) and Uq(g) are also discussed. Finally, the generalisation from quantum groups to general quantum Riemannian geometry is outlined. The semiclassical limit of the latter is a theory with generalised non-symmetric metric gµν obeying ∇µgνρ − ∇νgµρ = 0. 1