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47
Extensions of locally compact quantum groups and the bicrossed product construction
, 2001
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Classification of finitedimensional triangular Hopf algebras with the Chevalley property
 Mathematical Research Letters
"... Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose ..."
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Cited by 18 (9 self)
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Abstract. A fundamental problem in the theory of Hopf algebras is the classification and construction of finitedimensional quasitriangular Hopf algebras over C. Quasitriangular Hopf algebras constitute a very important class of Hopf algebras, introduced by Drinfeld. They are the Hopf algebras whose representations form a braided tensor category. However, this intriguing problem is extremely hard and is still widely open. Triangular Hopf algebras are the quasitriangular Hopf algebras whose representations form a symmetric tensor category. In that sense they are the closest to group algebras. The structure of triangular Hopf algebras is far from trivial, and yet is more tractable than that of general Hopf algebras, due to their proximity to groups. This makes triangular Hopf algebras an excellent testing ground for general Hopf algebraic ideas, methods and conjectures. A general classification of triangular Hopf algebras is not known yet. However, the problem was solved in the semisimple case, in the minimal triangular pointed case, and more generally for triangular Hopf algebras with the Chevalley property. In this paper we report on all of this, and explain in full details the mathematics and ideas involved in this theory. The classification in the semisimple case relies on Deligne’s theorem on Tannakian categories and on Movshev’s theory in an essential way. We explain Movshev’s theory in details, and refer to [G5] for a detailed discussion of the first aspect. We also discuss the existence of grouplike elements in quasitriangular semisimple Hopf algebras, and the representation theory of cotriangular semisimple Hopf algebras. We conclude the paper with a list of open problems; in particular with the question whether any finitedimensional triangular Hopf algebra over C has the Chevalley property. 1.
Nonsemiregular quantum groups coming from number theory
 COMMUN. MATH. PHYS
, 2002
"... In this paper, we study C∗algebraic quantum groups obtained through the bicrossed product construction. Examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semiregular: the crossed product of the quantum group acting on itse ..."
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Cited by 13 (6 self)
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In this paper, we study C∗algebraic quantum groups obtained through the bicrossed product construction. Examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semiregular: the crossed product of the quantum group acting on itself by translations does not contain any compact operator. We describe all corepresentations of these quantum groups and the associated universal C∗algebras. On the way, we provide several remarks on C∗algebraic properties of quantum groups and their actions.
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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Cited by 13 (1 self)
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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
On Group Theoretical Hopf Algebras and Exact Factorizations of Finite Groups
 J. of Algebra
, 2003
"... Abstract. We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the DijkgraafPasquierRoche quasiHopf algebra D ω (Σ), for some finite group Σ and some ω ∈ Z 3 (Σ, k ×). We show that semisimple Hopf algebras obtained as bicrossed products ..."
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Cited by 11 (4 self)
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Abstract. We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the DijkgraafPasquierRoche quasiHopf algebra D ω (Σ), for some finite group Σ and some ω ∈ Z 3 (Σ, k ×). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Σ are group theoretical. We also describe their Drinfeld double as a twisting of D ω (Σ), for an appropriate 3cocycle ω coming from the Kac exact sequence. 1.
On lowdimensional locally compact quantum groups
 PROCEEDINGS OF THE MEETING OF THEORETICAL PHYSICISTS AND MATHEMATICIANS, STRASBOURG, FEBRUARY 21  23, 2002., ED. L. VAINERMAN, IRMA LECTURES ON MATHEMATICS AND MATHEMATICAL PHYSICS, WALTER DE GRUYTER
, 2003
"... Continuing our research on extensions of locally compact quantum groups, we give a classification of all cocycle matched pairs of Lie algebras in small dimensions and prove that all of them can be exponentiated to cocycle matched pairs of Lie groups. Hence, all of them give rise to locally compact q ..."
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Cited by 10 (5 self)
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Continuing our research on extensions of locally compact quantum groups, we give a classification of all cocycle matched pairs of Lie algebras in small dimensions and prove that all of them can be exponentiated to cocycle matched pairs of Lie groups. Hence, all of them give rise to locally compact quantum groups by the cocycle bicrossed product construction. We also clarify the notion of an extension of locally compact quantum groups by relating it to the concept of a closed normal quantum subgroup and the quotient construction. Finally, we describe the infinitesimal objects of locally compact quantum quantum groups with 2 and 3 generatorsHopf ∗algebras and Lie bialgebras.
Multiparameter quantum function algebra at roots of 1
 759–780. RIGHT COIDEAL SUBALGEBRAS 13
"... In this paper we consider a multiparameter deformation Fϕ q [G] of the quantum function algebra associated to a simple algebraic group G. This deformation has been introduced by Reshetikhin ([R], cf. also [DKP1]) and is constructed from a skew endomorphism ϕ of the weight lattice of G. When ϕ is z ..."
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Cited by 8 (2 self)
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In this paper we consider a multiparameter deformation Fϕ q [G] of the quantum function algebra associated to a simple algebraic group G. This deformation has been introduced by Reshetikhin ([R], cf. also [DKP1]) and is constructed from a skew endomorphism ϕ of the weight lattice of G. When ϕ is zero we get the standard quantum group, that is the algebra studied by [HL123],
Examples of locally compact quantum groups through the bicrossed product construction
, 2000
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Hopf Algebra Extensions and Cohomology
"... Abstract. This is an expository paper on ‘abelian ’ extensions of (quasi) Hopf algebras, which can be managed by the abelian cohomology, with emphasis on the author’s recent results which are motivated by an exact sequence due to George Kac. The cohomology plays here an important role in constructi ..."
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Cited by 7 (0 self)
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Abstract. This is an expository paper on ‘abelian ’ extensions of (quasi) Hopf algebras, which can be managed by the abelian cohomology, with emphasis on the author’s recent results which are motivated by an exact sequence due to George Kac. The cohomology plays here an important role in constructing and classifying those extensions, and even their cocycle deformations. We see also a strong connection of Hopf algebra extensions arising from a (matched) pair of Lie algebras with Lie bialgebra extensions.
Double categories and quantum groupoids
, 2003
"... Abstract. We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now wellknown construction for Hopf algebras, first studied by G. I. Kac in the sixties. Our approach is based on the not ..."
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Cited by 6 (2 self)
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Abstract. We give the construction of a class of weak Hopf algebras (or quantum groupoids) associated to a matched pair of groupoids and certain cocycle data. This generalizes a now wellknown construction for Hopf algebras, first studied by G. I. Kac in the sixties. Our approach is based on the notion of double groupoids, as introduced by Ehresmann.