this paper we will discuss the combination of two classical notions of category theory, both treated extensively in [CWM]. One of these is the notion of a monad or triple on a category, which goes back to Godement [G] and was rst developed by Eilenberg, Moore, Beck and others. The other is that of a monoidal category or tensor category, which originates with Benabou [Be] and with Mac Lane's famous coherence theorem [MacL], and which pervades much of present day mathematics. For a monad S on a tensor category, there is a natural additional structure that one can impose, namely that of a comparison map S(X

To a Hecke symmetry R there associate a matrix bialgebra ER and a matrix Hopf algebra HR , which are called function rings on the matrix quantum semi-group and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum group are absolutely reducible and compute the integral on the function ring of the quantum group, i.e, on HR . Further, we show that the fusion coefficients of simple representations depend only on the rank of the symmetry, and give the explicit formula for the rank or 8-dim of HR-comodules. In the general case, we show that the quantum semi-group is "Zariski" dense in the quantum group. This enables us to study the semi-simplicity of the associated quantum group in some case. 1.

Abstract. This paper is a study of monoidal categories with duals where the tensor product need not be commutative. The motivating examples are categories of representations of Hopf algebras. We introduce the new notion of a spherical category. In the first section we prove a coherence theorem for a monoidal category with duals following [MacLane 1963]. In the second section we give the definition of a spherical category, and construct a natural quotient which is also spherical. In the

The significance of the 2-dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street in [JoS91]. Independently, Burroni has introduced a general notion of n-dimensional presentation in [Bur91] and he has shown that the equational logic of terms is a special case of 2-dimensional calculus. Here, we propose a combinatorial definition of 2-dimensional diagrams and a simple method for proving that certain monoidal categories are finitely 2-presentable. We illustrate the translation of terms into diagrams and we explain the change from groups to quantum groups in a purely syntactical way. This paper should serve as a reference for our future work on symbolic computation, including a theory of 2-dimensional rewriting and the design of software for interactive diagrammatic reasoning. New address: CNRS - Laboratoire de math'ematiques discr`etes, 163 avenue de Luminy - Case 930, 13288 Marseille Cedex 9, France. Email: lafont@lmd.univ-mrs.fr 2 1 FROM T...

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The starting point of the investigations described here was our discovery of a natural inner product on the ring K(n) ∗ BG, the n’th Morava K-theory of the classifying space of a finite group G. If n = 1 and G is a p-group then K(1) ∗ BG is essentially the same as R(G)/p (where R(G) is the complex representation ring of

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We study topological defect lines in two-dimensional rational conformal field theory. Continuous variation of the location of such a defect does not change the value of a correlator. Defects separating different phases of local CFTs with the same chiral symmetry are included in our discussion. We show how the resulting onedimensional phase boundaries can be used to extract symmetries and order-disorder dualities of the CFT. The case of central charge c = 4/5, for which there are two inequivalent world sheet phases corresponding to the tetra-critical Ising model and the critical three-states

Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.