相容、可裂和族Loday-代数

摘要：

给定一个非对称运算$\mathcal{O}$，我们首先构造了两个新的非对称运算$\mathcal{O}^{\mathrm{comp}}$和$\mathcal{O}^{\mathrm{Dend}}$。这些运算分别对研究相容和分裂Loday-代数是有用的。作为运算符$\mathcal{O}^{\mathrm{comp}}$的一个应用，我们证明了相容结合代数的上同调具有Gerstenhaber结构.给出了运算符$\mathcal{O}^\mathrm{Dend}$在树状代数上的应用，并推广到其他Loday-代数。最后，我们构造了另一个运算$\mathrm{Fam}(\mathcal{O}^Ω)^\mathrm{Dend}$来研究最近文献中引入的树状族代数。我们还定义和研究同伦树形族代数。

英文摘要：

Given a nonsymmetric operad $\mathcal{O}$, we first construct two new nonsymmetric operads $\mathcal{O}^{\mathrm{comp}}$ and $\mathcal{O}^{\mathrm{Dend}}$. These operads are respectively useful to study compatible and split Loday-algebras. As an application of the operad $\mathcal{O}^{\mathrm{comp}}$, we show that the cohomology of a compatible associative algebra carries a Gerstenhaber structure. We give an application of the operad $\mathcal{O}^\mathrm{Dend}$ to dendriform algebras and find generalizations to other Loday-algebras. In the end, we construct another operad $\mathrm{Fam}(\mathcal{O}^Ω)^\mathrm{Dend}$ to study dendriform-family algebras recently introduced in the literature. We also define and study homotopy dendriform-family algebras.

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