CLASSIFYING COALGEBRA SPLIT EXTENSIONS OF HOPF ALGEBRAS

Module Structure on Lie Powers and Natural Coalgebra-split Sub Hopf Algebras of Tensor Algebras

In this article, we investigate the functors from modules to modules that occur as the summands of tensor powers and the functors from modules to Hopf algebras that occur as natural coalgebra summands of tensor algebras. The main results provide some explicit natural coalgebra summands of tensor algebras. As a consequence, we obtain some decompositions of Lie powers over the general linear groups.

Hopf Algebra Extensions of Monogenic Hopf Algebras

William M. Singer has described a cohomology theory of connected Hopf algebras which classifies extensions of a cocommutative Hopf algebra by a commutative Hopf algebra in much the same way as the cohomology of groups classifies extensions of a group by an abelian group. We compute these cohomology groups for monogenic Hopf algebras, construct an action of the base ring on the cohomology groups...

Cocycle Deformations for Hopf Algebras with a Coalgebra Projection

In this paper we study Hopf algebras with a coalgebra projection A ∼= R#ξH and their deformations by an H-bilinear cocycle. If γ is a cocycle for A, then γ can be restricted to a cocycle γR for R, and A γ ∼= RR#ξγH. As examples, we consider liftings of B(V )#K[Γ] where Γ is a finite abelian group, V is a quantum plane and B(V ) is its Nichols algebra, and explicitly construct the cocycle which ...

On split exact sequences Hopf algebras

Galois extensions for coquasi-Hopf algebras

The notions of Galois and cleft extensions are generalized for coquasi-Hopf algebras. It is shown that such an extension over a coquasi-Hopf algebra is cleft if and only if it is Galois and has the normal basis property. A Schneider type theorem ([33]) is proven for coquasi-Hopf algebras with bijective antipode. As an application, we generalize Schauenburg’s bialgebroid construction for coquasi...