Equivariant one-parameter deformations of Lie triple systems

Cohomology and 1-parameter formal deformations of Hom-Lie triple systems

In this paper, we study the cohomology theory of Hom-Lie triple systems generalizing the Yamaguti cohomology theory of Lie triple systems. We introduce the central extension theory for Hom-Lie triple systems and show that there is a one-to-one correspondence between equivalent classes of central extensions of Hom-Lie triple systems and the third cohomology group. We develop the 1-parameter formal deformation theory of Hom-Lie triple systems, and prove that it is governed by the cohomology group.

Deformations of relative Rota-Baxter operators on Leibniz Triple Systems

In this paper, we introduce the cohomology theory of relative Rota-Baxter operators on Leibniz triple systems. We use the cohomological approach to study linear and formal deformations of relative Rota-Baxter operators. In particular, formal deformations and extendibility of order $n$ deformations of a relative Rota-Baxter operators are also characterized in terms of the cohomology theory. We also consider the relationship between cohomology of relative Rota-Baxter operators on Leibniz algebras and associated Leibniz triple systems.

拟斜李三系上变形映射的上同调

Cohomology for Deformation maps on quasi-twilled Lie triple systems

本文针对拟斜李三系上的多种算子，通过研究其两种变形映射，建立相应上同调，为研究李三系上的算子提供统一方法，恢复了几种算子的上同调。

On equivariant dendriform algebras

Dendriform algebras are certain associative algebras whose product splits into two binary operations and the associativity splits into three new identities. In this paper, we study finite group actions on dendriform algebras. We define equivariant cohomology for dendriform algebras equipped with finite group actions similar to the Bredon cohomology for topological $G$-spaces. We show that equivariant cohomology of such dendriform algebras controls equivariant one-parameter formal deformations.

李三系之间非阿贝尔嵌入张量的上同调与形变

Cohomology and deformations of nonabelian embedding tensors between Lie triple systems

本文引入李三系间非阿贝尔嵌入张量，证明其诱导3-莱布尼茨代数，构造控制形变的$L_{\infty}$-代数与上同调，并用于刻画无穷小形变及Nijenhuis元素。

Deformations of algebras defined by tilting bundles

In this paper we produce noncommutative algebras derived equivalent to deformations of schemes with tilting bundles. We do this in two settings, first proving that a tilting bundle on a scheme lifts to a tilting bundle on an infinitesimal deformations of that scheme and then expanding this result to $\mathbb{C}^*$-equivariant deformations over schemes with a good $\mathbb{C}^*$-action. In both these situations the endomorphism algebra of the lifted tilting bundle produces a deformation of the original endomorphism algebra, and this is a graded deformation in the $\mathbb{C}^*$-equivariant case…

Cohomology of Leibniz Triple Systems and its applications

In this paper, we introduce the first and third cohomology groups on Leibniz triple systems, which can be applied to extension theory and $1$-parameter formal deformation theory. Specifically, we investigate the central extension theory for Leibniz triple systems and show that there is a one-to-one correspondence between equivalent classes of central extensions of Leibniz triple systems and the third cohomology group. We study the $T^*$-extension of a Leibniz triple system and we determined that every even-dimensional quadratic Leibniz triple system $(\mathfrak{L},B)$ is isomorphic to a $T^*$-…

Localization algebras and deformations of Koszul algebras

We show that the center of a flat graded deformation of a standard Koszul algebra behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed-point set. In particular, the center acts by characters on the deformed standard modules, providing a "localization map." We construct a universal graded deformation, and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming the Koszul dual algebra. This is an algebraic version of a duality discovered by Goresky and MacPherson b…

带权的Hom-李三重系统上的O-算子

Weighted $\mathcal{O}$-operators on Hom-Lie triple systems

本文研究了Hom-李三重系统中带权O-算子的构造与上同调，提出了Hom-后李三重系统并展示了其与Hom-后李代数的关系。

受限李三系的上同调、扩张与形变

Cohomology, Extensions and Deformations of restricted Lie triple systems

本文构建了受限李三系的低阶上同调理论，定义了直到5阶的受限上链与上边界算子，并阐释了低阶上同调群在导子、模扩张、形式形变及系统扩张中的代数意义。