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Let H be a Hopf algebra of dimension pq over an algebraically closed field of characteristic 0, where p ≤ q are odd primes. Suppose that S is the antipode of H. If H is not semisimple, then S 4p = idH and Tr(S 2p) is an integer divisible by p 2. In particular, if dimH = p 2, we prove that H is isomorphic to a Taft algebra. We then complete the classification for the Hopf algebras of dimension p 2. 1

Abstract. By studying “points of the underlying quantum groups”of coquasitriangular Hopf (face) algebras, we construct ribbon categories for each lattice models without spectral parameter of both vertex and face type. Also, we give a classification of the braiding and the ribbon structure on quantized classical groups and modular tensor categories closely related to quantum SU(N)Linvariants of 3-manifolds. 1. introduction It is widely accepted that (co-)quasitriangular Hopf algebra is a good algebraic notion which expresses “quantum groups. ” For, example, each lattice model w of vertex type (and of face type) without spectral parameter naturally generates a coquasitriangular (CQT) Hopf (face) algebra, thanks to the FRT construction and the Hopf closure (or Hopf envelope) construction. The former construction assigns w to the CQT bialgebra (or face algebra) A(w) (cf. [30], [24], [31], [7]), while the latter construction assigns some CQT bialgebra (or face algebra) H to the CQT Hopf (face) algebra Hc(H) ([16], [14]). However, to give applications of CQT Hopf (face)

Abstract. We discuss some general results on finite-dimensional Hopf algebras over an algebraically closed field k of characteristic zero and then apply them to Hopf algebras H of dimension p 3 over k. There are 10 cases according to the group-like elements of H and H ∗. We show that in 8 of the 10 cases, it is possible to determine the structure of the Hopf algebra. We give also a partial classification of the quasitriangular Hopf algebras of dimension p 3 over k, after studying extensions of a group algebra of order p by a Taft algebra of dimension p 2. In particular, we prove that every ribbon Hopf algebra of dimension p 3 over k is either a group algebra or a Frobenius-Lusztig kernel. Finally, using some results from [1] and [4] on bounds for the dimension of the first term H1 in the coradical filtration of H, we give the complete classification of the quasitriangular Hopf algebras of dimension 27. 1.

Abstract. We construct finite-dimensional pointed Hopf algebras ur,s(G2) (i.e. restricted 2-parameter quantum groups) from the 2-parameter quantum group Ur,s(G2) defined in [HS], which turn out to be of Drinfel’d doubles, where a crucial point is to give a detailed combinatorial construction of the convex PBW-type Lyndon basis for type G2 in 2-parameter quantum version. After furnishing possible commutation relations among quantum root vectors, we show that the restricted quantum groups are ribbon Hopf algebras under certain conditions through determining their left and right integrals. Besides these, we determine all of the Hopf algebra isomorphisms of ur,s(G2) in terms of the description of the sets of its left (right) skew-primitive elements. 1.

Abstract. We outline a theory of quantum algebras and coalgebras and their resulting invariants of unoriented 1–1 tangles, knots and links, we outline a theory of oriented quantum algebras and coalgebras and their resulting invariants of oriented 1–1 tangles, knots and links, and we show how these algebras and coalgebras are related. Quasitriangular Hopf algebras are examples of quantum algebras and oriented quantum algebras; likewise coquasitriangular Hopf algebras are examples of quantum coalgebras and oriented quantum coalgebras.

Right integrals and invariants of three–manifolds

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We show how to construct, starting from a quasi-Hopf algebra, or quasi-quantum group, invariants of knots and links. In some cases, these invariants give rise to invariants of the three-manifolds obtained by surgery along these links. This happens for a finite-dimensional quasi-quantum group, whose definition involves a finite group G, and a 3-cocycle ω, which was first studied by Dijkgraaf, Pasquier and Roche. We treat this example in more detail, and argue that in this case the invariants agree with the partition function of the topological field theory of Dijkgraaf and Witten depending on the same data G, ω. CERN-TH 6360/92 Revised version February 1992 It is by now well established that there are deep connections between two-dimensional rational conformal field theories (RCFT), three-dimensional topological field theories (TFT), and quantum groups when q is a root of unity, see e.g. [1, 2, 3, 4, 5, 6, 7]. Some aspects of this