Hopf superquivers.
The authors follow some of the ideas of \textit{Y. Han} and \textit{D. Zhao} [J. Algebra 321, No. 12, 3668-3680 (2009; Zbl 1223.16018)], where a class of superalgebras is studied by means of superspecies. We recall that a supercoalgebra over a field \(K\) is a \(\mathbb Z_2\)-graded coalgebra \(C=C_0\oplus C_1\) satisfying some obvious conditions. Superalgebras, superbialgebras, and Hopf superalgebras are defined in a similar way. Following the results of Han-Zhao and the idea of \textit{C. Cibils} and \textit{M. Rosso} [J. Algebra 254, No. 2, 241-251 (2002; Zbl 1020.16025)], the authors develop a ``super version'' of Hopf quiver. They introduce the concepts of a superquiver and a Hopf superquiver and study the representations of pointed Hopf superalgebras in a superquiver setting. Among others, a super version of the well-known Gabriel type theorem and the Cartier-Gabriel decomposition theorem is proved in the paper. Interesting examples are also discussed. We recall that a quiver is a digraph \(Q=(Q_0,Q_1)\), that is, a directed multigraph with the set of vertices \(Q_0\) and the set \(Q_1\) of arrows (directed edges). By a superquiver the authors mean a quiver \(Q=(Q_0,Q_1)\) equipped with a parity map \(p\colon Q_1\to\mathbb Z_2=\{0,1\}\). One should note that superquivers are known in the literature as signed directed multigraphs introduced by \textit{F. Harary} [Mich. Math. J. 2, 143-146 (1954; Zbl 0056.42103)] and developed by \textit{T. Zaslavsky} [Discrete Appl. Math. 4, 47-74 (1982; Zbl 0476.05080)].
scientific article; zbMATH DE number 5911385
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