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, 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.
Bulacu et al. (Wed,) studied this question.