Cocycle Deformations for Liftings of Quantum Linear Spaces

Let $A$ be a Hopf algebra over a field $K$ of characteristic $0$ and suppose there is a coalgebra projection $pi$ 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 $R #_{\xi}H$ where $(R,\xi)$ is called a pre-bialgebra with cocycle in the category $_{H}^{H}mathcal{YD}$. The cocycle $\xi$ maps $R otimes R$ to $H$. Examples of this situation include the liftings of pointed Hopf algebras with abelian group of points $Gamma$ as classified by Andruskiewitsch and Schneider cite{ASannals}. One asks when such an $A$ can be twisted by a cocycle $gamma:Aotimes A ightarrow K$ to obtain a Radford biproduct. By results of Masuoka cite{Mas2, Mas1}, and Gr"{u}nenfelder and Mastnak cite{Grunenfelder-Mastnak}, this can always be done for the pointed liftings mentioned above. In a previous paper cite{ABM}, we showed that a natural candidate for a twisting cocycle is { $lambda circ \xi$} where $lambdain H^{ast}$ is a total integral for $H$ and $\xi$ 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 { $lambda circ \xi$}. In this note we show that in many cases this cocycle is exactly $lambdacirc\xi$ and give some further examples where this is not the case. As well we extend the cocycle computation to quantum linear spaces; there is no restriction on the dimension.

Data di pubblicazione: 2011
Titolo: Cocycle Deformations for Liftings of Quantum Linear Spaces
Autori: Ardizzoni, Alessandro; Beattie, M.; Menini, Claudia
Rivista: COMMUNICATIONS IN ALGEBRA
Keywords: Hopf algebra; coalgebra projection; cocycle twist; quantum linear space.
Abstract: Let $A$ be a Hopf algebra over a field $K$ of characteristic $0$ and suppose there is a coalgebra projection $pi$ 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 $R #_{\xi}H$ where $(R,\xi)$ is called a pre-bialgebra with cocycle in the category $_{H}^{H}mathcal{YD}$. The cocycle $\xi$ maps $R otimes R$ to $H$. Examples of this situation include the liftings of pointed Hopf algebras with abelian group of points $Gamma$ as classified by Andruskiewitsch and Schneider cite{ASannals}. One asks when such an $A$ can be twisted by a cocycle $gamma:Aotimes A ightarrow K$ to obtain a Radford biproduct. By results of Masuoka cite{Mas2, Mas1}, and Gr"{u}nenfelder and Mastnak cite{Grunenfelder-Mastnak}, this can always be done for the pointed liftings mentioned above. In a previous paper cite{ABM}, we showed that a natural candidate for a twisting cocycle is { $lambda circ \xi$} where $lambdain H^{ast}$ is a total integral for $H$ and $\xi$ 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 { $lambda circ \xi$}. In this note we show that in many cases this cocycle is exactly $lambdacirc\xi$ and give some further examples where this is not the case. As well we extend the cocycle computation to quantum linear spaces; there is no restriction on the dimension.
Digital Object Identifier (DOI): 10.1080/00927872.2011.616430
Handle: http://hdl.handle.net/11392/1464118
Appare nelle tipologie:03.1 Articolo su rivista

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