Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
Deformations and Cohomologies of Relative Rota-Baxter Operators on Lie Algebroids and Koszul-Vinberg Structures
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DOI：
10.3842/sigma.2022.054
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发表时间：
2021-08
期刊：
影响因子：
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通讯作者：
Mei-Lun Liu;Jiefeng Liu;Y. Sheng
中科院分区：
文献类型：
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作者：
Mei-Lun Liu;Jiefeng Liu;Y. Sheng
给定一个具有表示的李代数体，我们构造一个分级李代数，其 Maurer-Cartan 元素描述了李代数体上的相对 Rota-Baxter 算子。我们给出了相关Rota-Baxter算子的上同调，并根据该上同调理论研究了相关Rota-Baxter算子的无穷小变形以及n阶变形到n+1阶变形的可扩展性。我们还在向量丛的多重导数空间上构造了分级李代数，该向量丛的 Maurer-Cartan 元素表征了左对称代数胚。我们证明了李代数体上相关 Rota-Baxter 算子的控制分级李代数与左对称代数体的控制分级李代数之间存在同态。因此，从相关 Rota-Baxter 算子的上同调群到相关左对称代数体的变形上同调群存在自然同态。作为应用，我们给出了左对称代数体上Koszul-Vinberg结构的控制分级李代数和上同调理论。
Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer-Cartan elements characterize relative Rota-Baxter operators on Lie algebroids. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extendability of order n deformations to order n+1 deformations of relative Rota-Baxter operators in terms of this cohomology theory. We also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer-Cartan elements characterize left-symmetric algebroids. We show that there is a homomorphism from the controlling graded Lie algebra of relative Rota-Baxter operators on Lie algebroids to the controlling graded Lie algebra of left-symmetric algebroids. Consequently, there is a natural homomorphism from the cohomology groups of a relative Rota-Baxter operator to the deformation cohomology groups of the associated left-symmetric algebroid. As applications, we give the controlling graded Lie algebra and the cohomology theory of Koszul-Vinberg structures on left-symmetric algebroids.