1-Cycle Deformations for Yetter-Drinfeld Coalgebras

Abstract

For a quasi-Hopf algebra H, we study two types of 1-cycle deformations for a coalgebra C within the category of Yetter-Drinfeld modules over H, \({}_H^H{\mathcal YD}\). The two deformations produce C-comodule structures in \({}_H^H{\mathcal YD}\) and new coalgebra structures on C in \({}_H^H{\mathcal YD}\), respectively. We show that the isomorphism types of these structures are described by a 1-homology \({\mathcal H}^1_H(C, H_0)\) that we will introduce. Then we apply our results to the so called symplectic fermion quasi-Hopf algebras, algebras recently introduced by Farsad, Gainutdinov and Runkel.