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Classifying bicrossed products of Hopf algebras

Journal Contribution - Journal Article

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 H2(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) that can be defined between A and H. In the construction of H2(A,H) the key role is played by special elements of CoZ1(H,A)Ă—Aut 1 CoAlg(H), where CoZ1(H,A) is the group of unitary cocentral maps and Aut 1 CoAlg(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, 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) of Hopf algebra automorphisms is fully described.
Journal: Algebras and Representation Theory
ISSN: 1386-923X
Issue: 17
Pages: 227-264
Publication year:2014
Keywords:Bicrossed product, factorization problem, classification of Hopf algebras
  • Scopus Id: 84893647652