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38
On modules associated to coalgebraGalois extensions
 J. Algebra
, 1999
"... For a given entwining structure involving an algebra A, a coalgebra C, and an entwining map ψ: C ⊗A → A ⊗C, a category of right (A,C,ψ)modules is defined and its structure analysed. It is shown that in the case of the canonical entwining structure of a CGalois extension A of an algebra B this cate ..."
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Cited by 53 (14 self)
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For a given entwining structure involving an algebra A, a coalgebra C, and an entwining map ψ: C ⊗A → A ⊗C, a category of right (A,C,ψ)modules is defined and its structure analysed. It is shown that in the case of the canonical entwining structure of a CGalois extension A of an algebra B this category is equivalent to the category of left Bmodules if and only if A is faithfully flat as a left Bmodule. Left modules E and right modules Ē associated to a CGalois extension A of B are defined. These can be thought of as objects dual to fibre bundles with coalgebra C in the place of a structure group, and a fibre V. Crosssections of such associated modules are defined as module maps E → B or Ē → B. It is shown that they can be identified with suitably equivariant maps from the fibre to A. Also, it is shown that a CGalois extension is cleft if and only if A = B ⊗ C as left Bmodules and right Ccomodules. The relationship between modules E and Ē is studied in the case when V is finitedimensional and in the case when the canonical entwining map is bijective. 1.
Coalgebra extensions and algebra coextensions of Galois type, Commun. Algebra 27
, 1999
"... The notion of a coalgebraGalois extension is defined as a natural generalisation of a HopfGalois extension. It is shown that any coalgebraGalois extension induces a unique entwining map ψ compatible with the right coaction. For the dual notion of an algebraGalois coextension it is also proven th ..."
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Cited by 49 (15 self)
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The notion of a coalgebraGalois extension is defined as a natural generalisation of a HopfGalois extension. It is shown that any coalgebraGalois extension induces a unique entwining map ψ compatible with the right coaction. For the dual notion of an algebraGalois coextension it is also proven that there always exists a unique entwining structure compatible with the right action. 1
Quantum geometry of algebra factorisations and coalgebra bundles
 Commun. Math. Phys
, 2000
"... We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M2(C) = CZ2 · CZ2. We also further extend the coalgebra version of theory introduced previously, to include frame bundles and el ..."
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Cited by 34 (15 self)
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We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M2(C) = CZ2 · CZ2. We also further extend the coalgebra version of theory introduced previously, to include frame bundles and elements of Riemannian geometry. As an example, we construct qmonopoles on all the Podle´s quantum spheres S 2 q,s. 1.
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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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 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.
An Approach to Hopf Algebras via Frobenius Coordinates
, 1999
"... In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of a ..."
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Cited by 9 (4 self)
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In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FHalgebra H [25] and extend two recent theorems in [8]. We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in [10]. We study the Frobenius structure on an FHsubalgebra pair in Sections 5 and 6. In Section 8 we show that the quantum double of H is symmetric and unimodular.
Galois theory for bialgebroids, depth two and normal Hopf subalgebras
"... Abstract. We reduce certain proofs in [16, 11, 12] to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a lef ..."
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Cited by 9 (6 self)
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Abstract. We reduce certain proofs in [16, 11, 12] to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a left TGalois extension for some right finite projective left bialgebroid over some algebra R if and only if it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a HopfGalois extension. We characterize finite weak HopfGalois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.
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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Cited by 8 (1 self)
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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