Injective Hopf bimodules, cohomologies of infinite dimensional Hopf algebras and graded-commutativity of the Yoneda product. (Q1883059)

Let \(H\) be a Hopf algebra. A Hopf bimodule \(M\) is a bimodule and a bicomodule such that these structures are compatible. It is shown that the category of Hopf bimodules has enough injectives. Then the author shows that the cohomologies introduced by \textit{M. Gerstenhaber} and \textit{S. D. Schack} [in Ann. Math. (2) 78, 267-288 (1963; Zbl 0131.27302) and Proc. Natl. Acad. Sci. USA 87, No. 1, 478-481 (1990; Zbl 0695.16005)] and by \textit{C. Ospel} [in his thesis] coincide. This is obtained via an interpretation of these cohomologies as extension in appropiate categories. Using techniques introduced by \textit{S. Schwede} [J. Reine Angew. Math. 498, 153-172 (1998; Zbl 0923.16007)] it is shown that the Yoneda product of Hopf bimodule extensions of \(H\) by \(H\) is graded-commutative. In the last section, it is observed that the natural candidates for a graded Lie bracket for the described cohomologies which is compatible with the cup-product are trivial. In the Appendix, it is shown that their bracket is equivalent to the one constructed by \textit{M. A. Farinati} and \textit{A. L. Solotar} [Proc. Am. Math. Soc. 132, No. 10, 2859-2865 (2004; see Zbl 1063.16010 above)].