当前位置: X-MOL 学术Homol. Homotopy Appl. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Kapranov’s construction of $\operatorname{sh}$ Leibniz algebras
Homology, Homotopy and Applications ( IF 0.8 ) Pub Date : 2020-01-01 , DOI: 10.4310/hha.2020.v22.n1.a9
Zhuo Chen 1 , Zhangju Liu 2 , Maosong Xiang 3
Affiliation  

Motivated by Kapranov's discovery of an $L_\infty$ algebra structure on the tangent complex of a K\"ahler manifold and Chen-Sti\'enon-Xu's construction of a Leibniz$_\infty[1]$ algebra associated with a Lie pair, we find a general method to construct Leibniz$_\infty[1]$ algebras --- from a DG derivation $\mathscr{A} \xrightarrow{\delta} \Omega$ of a commutative differential graded algebra $\mathscr{A}$ valued in a DG $\mathscr{A}$-module $\Omega$. We prove that for any $\delta$-connection $\nabla$ on $\mathcal{B}$, the $\mathscr{A}$-dual of $\Omega$, there associates a Leibniz$_\infty[1]$ $\mathscr{A}$-algebra $(\mathcal{B},\{\mathcal{R}^\nabla_k\}_{k\geq 1})$. Moreover, this construction is canonical, i.e., the isomorphism class of $(\mathcal{B},\{\mathcal{R}^\nabla_k\}_{k\geq 1})$ only depends on the homotopy class of $\delta$.

中文翻译:

Kapranov 对 $\operatorname{sh}$ 莱布尼茨代数的构造

受 Kapranov 在 K\"ahler 流形的切复形上发现 $L_\infty$ 代数结构和 Chen-Sti\'enon-Xu 构建与李相关的 Leibniz$_\infty[1]$ 代数的启发对,我们找到了构造 Leibniz$_\infty[1]$ 代数的一般方法 --- 从交换微分分级代数 $\mathscr 的 DG 推导 $\mathscr{A} \xrightarrow{\delta} \Omega$ {A}$ 在 DG $\mathscr{A}$-module $\Omega$ 中的值。我们证明对于 $\mathcal{B}$ 上的任何 $\delta$-connection $\nabla$,$\mathscr {A}$-$\Omega$ 的对偶,有一个 Leibniz$_\infty[1]$ $\mathscr{A}$-代数 $(\mathcal{B},\{\mathcal{R}^\ nabla_k\}_{k\geq 1})$. 而且,这个构造是规范的,即 $(\mathcal{B},\{\mathcal{R}^\nabla_k\}_{k\ geq 1})$ 仅取决于 $\delta$ 的同伦类。
更新日期:2020-01-01
down
wechat
bug