原斜纹莱布尼茨代数中的形变映射
Deformation maps in proto-twilled Leibniz algebras
AI总结：
该研究在原斜纹莱布尼茨代数中引入形变映射以统一研究莱布尼茨代数上各类算子的上同调，构造了对应弯曲$L_\infty$-代数并给出相关算子的Maurer-Cartan刻画，还构建了控制算子及代数与算子同时形变的两个主控$L_\infty$-代数。
摘要：
本文旨在寻找一种统一方法来研究莱布尼茨代数上各类算子的上同调理论。为此，我们首先在原斜纹莱布尼茨代数中引入形变映射。这类映射推广了定义在莱布尼茨代数上的多种熟知算子（如同态、导子、交叉同态、Rota-Baxter算子、修正Rota-Baxter算子、扭Rota-Baxter算子、Reynolds算子等）以及李代数上的嵌入张量。我们定义了形变映射的上同调，统一了上述所有算子已有的上同调理论。随后我们构造了一个弯曲$L_\infty$-代数，其Maurer-Cartan元素恰好是给定原斜纹莱布尼茨代数中的形变映射。特别地，我们得到了莱布尼茨代数上修正Rota-Baxter算子、扭Rota-Baxter算子和Reynolds算子的Maurer-Cartan刻画。最后，给定一个原斜纹莱布尼茨代数和一个形变映射$r$，我们构造了两个主控$L_\infty$-代数，第一个控制算子$r$的形变，第二个控制原斜纹莱布尼茨代数与算子$r$的同时形变。
英文摘要：
This paper aims to find a unified approach to studying the cohomology theories of various operators on Leibniz algebras. We first introduce deformation maps in a proto-twilled Leibniz algebra to do this. Such maps generalize various well-known operators (such as homomorphisms, derivations, crossed homomorphisms, Rota-Baxter operators, modified Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators etc) defined on Leibniz algebras and embedding tensors on Lie algebras. We define the cohomology of a deformation map unifying the existing cohomologies of all the operators mentioned above. Then we construct a curved $L_\infty$-algebra whose Maurer-Cartan elements are precisely deformation maps in a given proto-twilled Leibniz algebra. In particular, we get the Maurer-Cartan characterizations of modified Rota-Baxter operators, twisted Rota-Baxter operators and Reynolds operators on a Leibniz algebra. Finally, given a proto-twilled Leibniz algebra and a deformation map $r$, we construct two governing $L_\infty$-algebras, the first one controls the deformations of the operator $r$ while the second one controls the simultaneous deformations of both the proto-twilled Leibniz algebra and the operator $r$.
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