1-Cycle Deformations for Yetter-Drinfeld Coalgebras

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 , HH YD H H Y D . The two deformations produce C -comodule structures in HH YD H H Y D and new coalgebra structures on C in HH YD H H Y D , respectively. We show that the isomorphism types of these structures are described by a 1-homology H¹H(C, H₀) H H 1 ( 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.