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On the computation of Hopf 2-cocycles with an example of diagonal type
Published online by Cambridge University Press: 24 June 2022
Abstract
We present a framework for the computation of the Hopf 2-cocycles involved in the deformations of Nichols algebras over semisimple Hopf algebras. We write down a recurrence formula and investigate the extent of the connection with invariant Hochschild cohomology in terms of exponentials. As an example, we present detailed computations leading to the explicit description of the Hopf 2-cocycles involved in the deformations of a Nichols algebra of Cartan type
$A_2$
with
$q=-1$
, a.k.a. the positive part of the small quantum group
$\mathfrak{u}^+_{\sqrt{-\text{1}}}(\mathfrak{sl}_3)$
. We show that these cocycles are generically pure, that is they are not cohomologous to exponentials of Hochschild 2-cocycles.
Keywords
MSC classification
- Type
- Research Article
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- Copyright
- © The Author(s), 2022. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust
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