Extension theory for connected Hopf algebras

Hopf cohomology vanishing via approximation by Hochschild cohomology

“…The equivalence classes of those cleft extensions which are associated as above to a fixed Singer pair (H, K, , ρ) form an abelian group, Opext (H, K, , ρ); this is isomorphic to some cohomology group H 2 (H, K, , ρ). See [Si;H;M2] and Section 5 below.…”

“…Theorem 3.1 is the starting point of our new approach, given in Section 3, to Takeuchi's and Sullivan's Theorems. The same idea will be used in Section 5 to prove vanishing of the cohomology associated to an abelian matched pair [Si;T3] (or a Singer pair) of Hopf algebras.…”

Graded contractions and bicrossproduct structure of deformed inhomogeneous algebras

“…For instance, (A.5) is automatically satisfied, since U(so ω 2 ,...,ω N (N)) is undeformed and hence co-commutative and U λ (T N ) is abelian. This case, specially relevant here, was discussed in [18]. Then, the bicrossproduct structure of U λ (iso ω 2 ,...,ω N (N)) may be stated in the form of the following Theorem.…”

“…It is then natural to ask: is there a similar pattern for the contracted deformations i.e., for the Hopf algebra deformations of contracted simple Lie algebras? The analogue of the semidirect product is an example of the bicrossproduct of Hopf algebras, introduced by Majid [17] (see also [18,19]). The aim of this paper is to show that all deformed algebras in the affine ¶ CK family iso ω 2 ,...,ω N (N) have indeed a bicrossproduct structure, as is the case of the κ-Poincaré [21].…”

Connected Hopf algebras of dimensionp2

“…In [5], all graded cocommutative connected Hopf algebras of dimension less than or equal to p 3 are classified by using W.M. Singer's theory of extensions of connected Hopf algebras [13]. In this paper, we classify all connected Hopf algebras of dimension p 2 over k. We use the theories of restricted Lie algebras and…”