Hopf group-coalgebras (Q1612113)

Given a group \(\pi\), a Hopf \(\pi\)-coalgebra is a family \(H=\{H_\alpha\}_{\alpha\in\pi}\) of algebras (over a field \(k\)) endowed with a comultiplication \(\Delta=\{\Delta_{\alpha,\beta}\colon H_{\alpha\beta}\to H_\alpha\otimes H_\beta\}_{\alpha,\beta\in\pi}\), a counit \(\varepsilon\colon H_1\to k\) and an antipode \(S=\{S_\alpha\colon H_\alpha\to H_{\alpha^{-1}}\}_{\alpha\in\pi}\) satisfying coassociativity of \(\Delta\), counitary property for \(\varepsilon\) and \(\Delta_{1,1}\), \(\varepsilon\) and \(\Delta_{\alpha,\beta}\) are algebra maps for every \(\alpha,\beta\in\pi\), and \(S_\alpha\) is the inverse of the identity of \(H_{\alpha^{-1}}\) with respect to the convolution product. This notion was introduced by \textit{V. Turaev} [Homotopy field theory in dimension 3 and crossed group-categories, preprint, \url{arXiv:math.GT/0005291}]. Hopf \(\pi\)-comodules and rational \(\pi\)-graded modules are defined in a similar way. This paper sets the algebraic properties of Hopf \(\pi\)-coalgebras. The one-to-one correspondence between \(\pi\)-comodules and rational \(\pi\)-graded modules over the dual and the fundamental theorem of Hopf modules (\(M\cong M^{co H}\otimes H\)) are proved, extending well known results of \textit{M. E. Sweedler} [Hopf algebras, New York, W. A. Benjamin (1969; Zbl 0194.32901), Theorem 2.1.3] and \textit{R. G. Larson} and \textit{M. E. Sweedler} [Am. J. Math. 91, 75--94 (1969; Zbl 0179.05803), Proposition 1]. In the finite type case (i.e., \(H_\alpha\) is finite dimensional for all \(\alpha\in\pi\)) the space of left (resp. right) \(\pi\)-integrals is proved to be one dimensional, generalizing more of Sweedler's results, and they are related with distinguished \(\pi\)-grouplike elements as in [\textit{D. E. Radford}, J. Algebra 163, No. 3, 583--622 (1994; Zbl 0801.16039), Theorem 3]. Bounds on the order of the antipode are also provided. Semisimplicity and cosemisimplicity are studied too; a Hopf \(\pi\)-coalgebra is semisimple if and only if the degree \(1\) part is semisimple; a dual Maschke theorem is proved for cosemisimple Hopf \(\pi\)-coalgebras, and some consequences characterizing cosemisimplicity are derived. Finally, the algebraic structure of quasitriangular Hopf \(\pi\)-coalgebras is established, including the existence of \(\pi\)-traces for a finite type unimodular ribbon Hopf \(\pi\)-coalgebra.