Mathematical Research Letters

Volume 22 (2015)

Number 1

Deformation by cocycles of pointed Hopf algebras over non-abelian groups

Pages: 59 – 92

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n1.a5

Authors

Gastón Andrés García (Departamento de Matemática, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, CONICET, La Plata, Argentina)

Mitja Mastnak (Department of Mathematics, C.S. Saint Mary’s University, Halifax, Nova Scotia, Canada)

Abstract

We explore a method for explicitly constructing multiplicative $2$-cocycles for bosonizations of Nichols algebras $\mathfrak{B}(V)$ over Hopf algebras $H$. These cocycles arise as liftings of $H$-invariant linear functionals on $V \otimes V$ and give a formula for deforming braided-commutator-type relations. Using this construction, we show that all known finite-dimensional pointed Hopf algebras over the dihedral groups $\mathbb{D}_m$ with $m = 4t \geq 12 \,$, over the symmetric group $\mathbb{S}_3$, and some families over $\mathbb{S}_4$ are cocycle deformations of bosonizations of Nichols algebras.

2010 Mathematics Subject Classification

16T05

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