Dendriform-Nijenhuis bialgebras revisited
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In order to construct Nijenhuis algebra, two types of \linebreak dendriform-Nijenhuis (abbr. dN) bialgebras were introduced by Leroux in [Int. J. Math. Math. Sci. 49-52 (2004), 2595--2615], and then studied by Peng et al. in [J. Algebra 575 (2021), 78--126]. In this paper, we focus on the representations of dN-bialgebras. We find that these two types of dN-bialgebras correspond to left and right representations, respectively. Based on this, we refer to them as left and right dN-bialgebras. Further corresponding left and right representations can be explained by left and right dN-biproduct bialgebras, respectively. The fact that left and right dN-bialgebras possess the same dN-associative Yang--Baxter equations is discovered. At last, classifications of quasitriangular left (resp., right) dN-bialgebras of low dimensions are given.
Abstract
In order to construct Nijenhuis algebra, two types of \linebreak dendriform-Nijenhuis (abbr. dN) bialgebras were introduced by Leroux in [Int. J. Math. Math. Sci. 49-52 (2004), 2595--2615], and then studied by Peng et al. in [J. Algebra 575 (2021), 78--126]. In this paper, we focus on the representations of dN-bialgebras. We find that these two types of dN-bialgebras correspond to left and right representations, respectively. Based on this, we refer to them as left and right dN-bialgebras. Further corresponding left and right representations can be explained by left and right dN-biproduct bialgebras, respectively. The fact that left and right dN-bialgebras possess the same dN-associative Yang--Baxter equations is discovered. At last, classifications of quasitriangular left (resp., right) dN-bialgebras of low dimensions are given.