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18
From subfactors to categories and topology III. Triangulation invariants of 3manifolds and Morita equivalence of tensor categories
 In preparation
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Nonsemisimple Hopf algebras of dimension p 2
 J. Algebra
"... 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 isomo ..."
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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
Coribbon Hopf (face) algebras generated by lattice models
 J. Algebra
"... 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 quantiz ..."
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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 3manifolds. 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)
Convex PBWtype Lyndon basis and restricted twoparameter quantum groups of type B
, 2007
"... Abstract. We construct finitedimensional pointed Hopf algebras ur,s(G2) (i.e. restricted 2parameter quantum groups) from the 2parameter 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 ..."
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Abstract. We construct finitedimensional pointed Hopf algebras ur,s(G2) (i.e. restricted 2parameter quantum groups) from the 2parameter 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 PBWtype Lyndon basis for type G2 in 2parameter 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) skewprimitive elements. 1.
Convex PBWtype Lyndon bases and restricted twoparameter quantum groups of type G2
 Pacific J. Math
, 2009
"... Abstract. We construct convex PBWtype Lyndon bases for twoparameter quantum groups Ur,s(so2n+1) with detailed commutation relations. These serve as prerequisites of all discussions later. We show that the restricted twoparameter quantum groups ur,s(so2n+1) (r, s are roots of unity) are of Drinfeld ..."
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Abstract. We construct convex PBWtype Lyndon bases for twoparameter quantum groups Ur,s(so2n+1) with detailed commutation relations. These serve as prerequisites of all discussions later. We show that the restricted twoparameter quantum groups ur,s(so2n+1) (r, s are roots of unity) are of Drinfeld double under a certain condition. All of isomorphisms of ur,s(so2n+1), as well as ur,s(sln) are determined. Finally, necessary and sufficient conditions for ur,s(so2n+1) to be a ribbon Hopf algebra are given by describing the left and right integrals. 1.
On Hopf algebras of dimension p 3
 Tsukuba J. Math
"... Abstract. We discuss some general results on finitedimensional 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 grouplike elements of H and H ∗. We show that in 8 of the 10 ..."
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Abstract. We discuss some general results on finitedimensional 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 grouplike 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 FrobeniusLusztig 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.
Representations Of A Class Of Drinfeld’s Doubles
"... In 1971, Taft constructed an n2dimensional Hopf algebraAn(q), which is nonsemisimple. Before quantum groups appeared, there had been few examples of Hopf algebras except group algebras, the envelope algebras of Lie algebras, and their finite duals. In 1993, Kauffman and Radford proved that the Dri ..."
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In 1971, Taft constructed an n2dimensional Hopf algebraAn(q), which is nonsemisimple. Before quantum groups appeared, there had been few examples of Hopf algebras except group algebras, the envelope algebras of Lie algebras, and their finite duals. In 1993, Kauffman and Radford proved that the Drinfeld double (D(An(q), R) of An(q)
On Quantum Algebras and Coalgebras, Oriented Quantum Algebras and Coalgebras, Invariants of 1–1 Tangles, Knots and Links
"... 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 a ..."
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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.