Symmetric cohomology and symmetric Hochschild cohomology of cocommutative Hopf algebras (Q6637174)

The authors introduce symmetric cohomology for cocommutative Hopf algebras, by generalizing the symmetric cohomology of groups defined by \textit{M. D. Staic} in [J. Algebra 322, No. 4, 1360--1378 (2009; Zbl 1172.81014)], and symmetric Hochschild cohomology for cocommutative Hopf algebras. These cohomologies are defined by the subcomplexes of the standard non-homogeneous complexes (defining the respective non-symmetric cohomologies) that are fixed by an action of the symmetric group. Moreover, the authors provide an isomorphism between the symmetric cohomology and the symmetric Hochschild cohomology of cocommutative Hopf algebras as a symmetric version of the classical result about cohomology of groups obtained by \textit{S. Eilenberg} and \textit{S. MacLane} in [Ann. Math. (2) 48, 51--78 (1947; Zbl 0029.34001)] that was generalized to the case of Hopf algebras by \textit{V. Ginzburg} and \textit{S. Kumar} in [Duke Math. J. 69, No. 1, 179--198 (1993; Zbl 0774.17013)].