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EnglishLet H be a Hopf algebra over a field K of characteristic 0 and let A be a bialgebra or Hopf algebra such that H is isomorphic to a sub-Hopf algebra of A and there is an H-bilinear coalgebra projection π from A to H which splits the inclusion. Then A is isomorphic to R#ξH where R is the pre-bialgebra of coinvariants. In this paper we study the deformations of A 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 is isomorphic to R^γR#ξγ H. 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 twists the Radford biproduct into the lifting.
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| Data di pubblicazione: | 2010 | |
| Titolo: | Cocycle Deformations for Hopf Algebras with a Coalgebra Projection | |
| Autori: | ARDIZZONI A.; BEATTIE M.; MENINI C. | |
| Rivista: | JOURNAL OF ALGEBRA | |
| Parole Chiave: | Hopf algebra; Coalgebra projection; Radford biproduct; Cocycle twist | |
| Abstract: | Let H be a Hopf algebra over a field K of characteristic 0 and let A be a bialgebra or Hopf algebra such that H is isomorphic to a sub-Hopf algebra of A and there is an H-bilinear coalgebra projection π from A to H which splits the inclusion. Then A is isomorphic to R#ξH where R is the pre-bialgebra of coinvariants. In this paper we study the deformations of A 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 is isomorphic to R^γR#ξγ H. 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 twists the Radford biproduct into the lifting. | |
| Digital Object Identifier (DOI): | 10.1016/j.jalgebra.2010.04.027 | |
| Handle: | http://hdl.handle.net/11392/1403925 | |
| Appare nelle tipologie: | 03.1 Articolo su rivista |
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