论文标题
$ l_ \ infty $ - 结构和兼容$ \ MATHCAL {O} $ - 运算符和兼容的树突状代数
$L_\infty$-structures and cohomology theory of compatible $\mathcal {O}$-operators and compatible dendriform algebras
论文作者
论文摘要
$ \ MATHCAL {O} $ - 运算符的概念是在bimodule上存在的cossotiative代数的情况下对Rota-baxter操作员的概括。兼容$ \ MATHCAL {O} $ - 运算符是一对，由两个$ \ Mathcal {O} $ - 运算符组成，可满足兼容性关系。兼容$ \ MATHCAL {O} $ - 运算符代数是一个代数，以及biModule和兼容的$ \ Mathcal {O} $ - 运算符。在本文中，我们构建了一个分级的谎言代数和一个$ l_ \ infty $ -Algebra，分别表征了兼容$ \ Mathcal {O} $ - 操作员和兼容$ \ Mathcal {O} $ - 运营商代数为Maurer-Cartan元素。使用这些特征，我们将这些结构的共同体学定义为应用，我们研究了兼容$ \ Mathcal {O} $的形式变形 - 运算符和兼容$ \ Mathcal {O} $ - 运算符 - 操作员代数。最后，我们考虑了对兼容树突形代数的简要研究研究，并找到了它们与兼容的关联代数和兼容$ \ Mathcal {O} $ - 运算符的共同体的关系。
The notion of $\mathcal{O}$-operator is a generalization of the Rota-Baxter operator in the presence of a bimodule over an associative algebra. A compatible $\mathcal{O}$-operator is a pair consisting of two $\mathcal{O}$-operators satisfying a compatibility relation. A compatible $\mathcal{O}$-operator algebra is an algebra together with a bimodule and a compatible $\mathcal{O}$-operator. In this paper, we construct a graded Lie algebra and an $L_\infty$-algebra that respectively characterize compatible $\mathcal{O}$-operators and compatible $\mathcal{O}$-operator algebras as Maurer-Cartan elements. Using these characterizations, we define cohomology of these structures and as applications, we study formal deformations of compatible $\mathcal{O}$-operators and compatible $\mathcal{O}$-operator algebras. Finally, we consider a brief cohomological study of compatible dendriform algebras and find their relationship with the cohomology of compatible associative algebras and compatible $\mathcal{O}$-operators.
