Cocycle Deformations for Liftings of Quantum Linear Spaces

Abstract

Let A be a Hopf algebra over a field K of characteristic 0, and suppose there is a coalgebra projection π from A to a sub-Hopf algebra H that splits the inclusion. If the projection is H-bilinear, then A is isomorphic to a biproduct R#_ξ H where (R, ξ) is called a pre-bialgebra with cocycle in the category . The cocycle ξ maps R ⊗ R to H. Examples of this situation include the liftings of pointed Hopf algebras with abelian group of points Γ as classified by Andruskiewitsch and Schneider [1]. One asks when such an A can be twisted by a cocycle γ: A ⊗ A → K to obtain a Radford biproduct. By results of Masuoka [12, 13], and Grünenfelder and Mastnak [10], this can always be done for the pointed liftings mentioned above.

In a previous article [3], we showed that a natural candidate for a twisting cocycle is λ ○ ξ where λ ∈H* is a total integral for H and ξ is as above. We also computed the twisting cocycle explicitly for liftings of a quantum linear plane and found some examples where the twisting cocycle we computed was different from λ ○ ξ. In this note, we show that in many cases this cocycle is exactly λ ○ ξ and give some further examples where this is not the case. Also, we extend the cocycle computation to quantum linear spaces; there is no restriction on the dimension.

Key Words:

• Cocycle deformations
• Hopf algebras
• Liftings
• Quantum linear spaces

2000 Mathematics Subject Classification:

• 16W30
• 16S40

ACKNOWLEDGMENT

Thanks to the referee for his/her careful reading of this note.

M. Beattie's research was supported by an NSERC Discovery Grant. She would like to thank the University of Ferrara for their hospitality during her visit in 2010. This article was written while A. Ardizzoni and C. Menini were members of G.N.S.A.G.A. with partial financial support from M.I.U.R. (PRIN 2007).

Notes

Communicated by S. Montgomery.

Dedicated to Mia Cohen on the occasion of her retirement.