Algebras and Representation Theory

, Volume 17, Issue 1, pp 227–264

Classifying Bicrossed Products of Hopf Algebras

Article

Abstract

Let A and H be two Hopf algebras. We shall classify up to an isomorphism that stabilizes A all Hopf algebras E that factorize through A and H by a cohomological type object \({\mathcal H}^{2} (A, H)\). Equivalently, we classify up to a left A-linear Hopf algebra isomorphism, the set of all bicrossed products A ⋈ H associated to all possible matched pairs of Hopf algebras \((A, H, \triangleleft, \triangleright)\) that can be defined between A and H. In the construction of \({\mathcal H}^{2} (A, H)\) the key role is played by special elements of \(CoZ^{1} (H, A) \times {\rm Aut}\,_{\rm CoAlg}^1 (H)\), where CoZ1 (H, A) is the group of unitary cocentral maps and \({\rm Aut}\,_{\rm CoAlg}^1 (H)\) is the group of unitary automorphisms of the coalgebra H. Among several applications and examples, all bicrossed products H4 ⋈ k[Cn] are described by generators and relations and classified: they are quantum groups at roots of unity H4n, ω which are classified by pure arithmetic properties of the ring ℤn. The Dirichlet’s theorem on primes is used to count the number of types of isomorphisms of this family of 4n-dimensional quantum groups. As a consequence of our approach the group Aut Hopf(H4n, ω) of Hopf algebra automorphisms is fully described.

Keywords

Bicrossed product Factorization problem Classification of Hopf algebras 

Mathematics Subject Classifications (2010)

16T05 16S40 

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Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.Faculty of EngineeringVrije Universiteit BrusselBrusselsBelgium
  2. 2.Faculty of Mathematics and Computer ScienceUniversity of BucharestBucharest 1Romania

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