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Extensions of locally compact quantum groups and the bicrossed product construction
, 2001
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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
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.
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 8 (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.
COHOMOLOGY OF ABELIAN MATCHED PAIRS AND THE KAC SEQUENCE
, 2002
"... Abstract. The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general ab ..."
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Cited by 3 (3 self)
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Abstract. The purpose of this paper is to introduce a cohomology theory for abelian matched pairs of Hopf algebras and to explore its relationship to Sweedler cohomology, to Singer cohomology and to extension theory. An exact sequence connecting these cohomology theories is obtained for a general abelian matched pair of Hopf algebras, generalizing those of Kac and Masuoka for matched pairs of finite groups and finite dimensional Lie algebras. The morphisms in the low degree part of this sequence are given explicitly, enabling concrete computations. In this paper we discuss various cohomology theories for Hopf algebras and their relation to extension theory. It is natural to think of building new algebraic objects from simpler structures, or to get information about the structure of complicated objects by
Hopf powers and orders of some bismash products
 J. Pure and Applied Algebra
"... Let H be a Hopf algebra over the field k. We study the nth Hopf power map of H, a linear endomorphism [n]: H → H. If H = kG is a group algebra, then [n] is the linear extension of the nth power map on G. If H is commutative, and thus represents an affine group scheme G, the nth Hopf power map rep ..."
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Cited by 2 (1 self)
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Let H be a Hopf algebra over the field k. We study the nth Hopf power map of H, a linear endomorphism [n]: H → H. If H = kG is a group algebra, then [n] is the linear extension of the nth power map on G. If H is commutative, and thus represents an affine group scheme G, the nth Hopf power map represents the
Hopf Algebra Extensions and Monoidal Categories
, 2002
"... Tannaka reconstruction provides a close link between monoidal categories and (quasi)Hopf algebras. We discuss some applications of the ideas of Tannaka reconstruction to the theory of Hopf algebra extensions, based on the following construction: For certain inclusions of a Hopf algebra into a coq ..."
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Cited by 1 (1 self)
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Tannaka reconstruction provides a close link between monoidal categories and (quasi)Hopf algebras. We discuss some applications of the ideas of Tannaka reconstruction to the theory of Hopf algebra extensions, based on the following construction: For certain inclusions of a Hopf algebra into a coquasibialgebra one can consider a natural monoidal category consisting of Hopf modules, and one can reconstruct a new coquasibialgebra from that monoidal category.
REPRESENTATIONS OF SOME HOPF ALGEBRAS ASSOCIATED TO THE SYMMETRIC GROUP Sn
, 2007
"... In this paper we study the representations of two semisimple Hopf algebras related to the symmetric group Sn, namely the bismash products Hn = k Cn #kSn−1 and its dual Jn = k Sn−1 #kCn = (Hn) ∗ , where k is an algebraically closed field of ..."
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In this paper we study the representations of two semisimple Hopf algebras related to the symmetric group Sn, namely the bismash products Hn = k Cn #kSn−1 and its dual Jn = k Sn−1 #kCn = (Hn) ∗ , where k is an algebraically closed field of
doi:10.1006/jabr.2002.9145 Hopf Algebra Extensions Arising from SemiDirect Products of Groups
, 2001
"... The aim of this paper is to study Hopf algebra extensions arising from semidirect products of groups in terms of group cohomology. This enables us to compute and describe explicitly some groups of Hopf algebra extensions. 2002 Elsevier Science (USA) The Hopf algebra extensions arising from a semi ..."
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The aim of this paper is to study Hopf algebra extensions arising from semidirect products of groups in terms of group cohomology. This enables us to compute and describe explicitly some groups of Hopf algebra extensions. 2002 Elsevier Science (USA) The Hopf algebra extensions arising from a semidirect product of groups is a special case of Hopf algebra extensions arising from a matched pair of groups. Such a construction was first considered by G. I. Kac in the 1960s (see [Kac]). He established an exact sequence (now called Kac sequence)