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Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... 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 equivalenc ..."
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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.
The Classification of Semisimple Hopf Algebras of dimension 16
 J. of Algebra
"... Abstract. In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple noncommutative Hopf algebras of dimension 16. Moreover, we prove that noncommutative semisimple Hopf algebras of di ..."
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Abstract. In this paper we completely classify nontrivial semisimple Hopf algebras of dimension 16. We also compute all the possible structures of the Grothendieck ring of semisimple noncommutative Hopf algebras of dimension 16. Moreover, we prove that noncommutative semisimple Hopf algebras of dimension p n, p is prime, cannot have a cyclic group of grouplikes. 1. Introduction. Recently various classification results were obtained for finitedimensional semisimple Hopf algebras over an algebraically closed field of characteristic 0. The smallest dimension, for which the question was still open, was 16. In this paper we completely classify all nontrivial (i.e. noncommutative and noncocommutative) Hopf algebras of dimension 16. Moreover, we consider all
EtingofKazhdan Quantization of Lie Superbialgebras
 Adv. Math
"... Abstract. For every semisimple Lie algebra gone can construct the DrinfeldJimbo algebra UDJ h (g). This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of UDJ h (g), Drinfeld used the KZequations to construct a quasiHopf algebra Ag. H ..."
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Abstract. For every semisimple Lie algebra gone can construct the DrinfeldJimbo algebra UDJ h (g). This algebra is a deformation Hopf algebra defined by generators and relations. To study the representation theory of UDJ h (g), Drinfeld used the KZequations to construct a quasiHopf algebra Ag. He proved that particular categories of modules over the algebras UDJ h (g) and Ag are tensor equivalent. Analogous constructions of the algebras UDJ h (g) and Ag exist in the case when g is a Lie superalgebra of type AG. However, Drinfeld’s proof of the above equivalence of categories does not generalize to Lie superalgebras. In this paper, we will discuss an alternate proof for Lie superalgebras of type AG. Our proof utilizes the EtingofKazhdan quantization of Lie (super)bialgebras. It should be mentioned that the above equivalence is very useful. For example, it has been used in knot theory to relate quantum group invariants and the Kontsevich integral. 1.
An Australian conspectus of higher categories

, 2004
"... Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional wo ..."
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Cited by 6 (0 self)
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Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higherdimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences
FINITE DIMENSIONAL QUASIHOPF ALGEBRAS WITH RADICAL OF CODIMENSION 2
, 2003
"... It is shown in [EO], Proposition 2.17, that a finite dimensional quasiHopf algebra with radical of codimension 1 is semisimple and 1dimensional. On the other hand, there exist quasiHopf (in fact, Hopf) algebras, whose radical has codimension 2. Namely, it is known [N] that these are exactly the N ..."
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It is shown in [EO], Proposition 2.17, that a finite dimensional quasiHopf algebra with radical of codimension 1 is semisimple and 1dimensional. On the other hand, there exist quasiHopf (in fact, Hopf) algebras, whose radical has codimension 2. Namely, it is known [N] that these are exactly the Nichols Hopf algebras H2n of
Some remarks on quantized Lie superalgebras of classical type
 Preprint. LINK INVARIANTS FROM LIE SUPERALGEBRAS 23
"... Abstract. In this paper we use the EtingofKazhdan quantization of Lie bisuperalgebras to investigate some interesting questions related to DrinfeldJimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the DJ type superalgebra associated to a Lie supera ..."
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Abstract. In this paper we use the EtingofKazhdan quantization of Lie bisuperalgebras to investigate some interesting questions related to DrinfeldJimbo type superalgebra associated to a Lie superalgebra of classical type. It has been shown that the DJ type superalgebra associated to a Lie superalgebra of type AG, with the distinguished Cartan matrix, is isomorphic to the EK quantization of the Lie superalgebra. The first main result in the present paper is to extend this to arbitrary Cartan matrices. This paper also contains two other main results: 1) a theorem stating that all highest weight modules of a Lie superalgebra of type AG can be deformed to modules over the corresponding DJ type superalgebra and 2) a super version of the DrinfeldKohno Theorem. 1.
LIFTINGS OF GRADED QUASIHOPF ALGEBRAS WITH RADICAL OF PRIME CODIMENSION
, 2004
"... Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasiHopf algebras over C, whose radical has codimension p. In [EG1],[EG2] we completely describe the set RG(p). Namely, we show that for p> 2, RG(p) consists of the quasiHopf algebras A(q) ..."
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Let p be a prime, and let RG(p) denote the set of equivalence classes of radically graded finite dimensional quasiHopf algebras over C, whose radical has codimension p. In [EG1],[EG2] we completely describe the set RG(p). Namely, we show that for p> 2, RG(p) consists of the quasiHopf algebras A(q) constructed in [G]
On Hopf algebras and their generalizations
 Comm. Algebra
, 2007
"... Abstract. We survey Hopf algebras and their generalizations. In particular, we compare and contrast three wellstudied generalizations (quasiHopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (hopfish algebras and Hopf monads). Each of these notions was originally introduce ..."
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Abstract. We survey Hopf algebras and their generalizations. In particular, we compare and contrast three wellstudied generalizations (quasiHopf algebras, weak Hopf algebras, and Hopf algebroids), and two newer ones (hopfish algebras and Hopf monads). Each of these notions was originally introduced for a specific purpose within a particular context; our discussion favors applicability to the theory of dynamical quantum groups. Throughout the note, we provide several definitions and examples in order to make this exposition accessible to readers with differing backgrounds.
ON RADICALLY GRADED FINITE DIMENSIONAL QUASIHOPF ALGEBRAS
, 2009
"... Abstract. In this paper we continue the structure theory of finite dimensional quasiHopf algebras started in [EG] and [G]. First, we completely describe the class of radically graded finite dimensional quasiHopf algebras over C, whose radical has prime codimension. As a corollary we obtain that if ..."
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Abstract. In this paper we continue the structure theory of finite dimensional quasiHopf algebras started in [EG] and [G]. First, we completely describe the class of radically graded finite dimensional quasiHopf algebras over C, whose radical has prime codimension. As a corollary we obtain that if p> 2 is a prime then any finite tensor category over C with exactly p simple objects which are all invertible must have FrobeniusPerron dimension p N, N = 1, 2,3, 4, 5 or 7. Second, we construct new examples of finite dimensional quasiHopf algebras which are not twist equivalent to a Hopf algebra. For instance, to every finite dimensional simple Lie algebra g and an odd integer n, coprime to 3 if g = G2, we attach a quasiHopf algebra of dimension n dimg. 1.