Kapranov's Construction of Homotopy Leibniz Algebras

Zhuo Chen Tsinghua University Zhangju Liu Peking University MAOSONG XIANG Peking University

Quantum Algebra mathscidoc:1809.29001

Motivated by Kapranov's discovery of an $L_\infty$ algebra structure on the tangent complex of a K\"{a}hler manifold and Chen-Sti\'{e}non-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 $\A$-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 $\delta$.
homotopy Leibniz algebra
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@inproceedings{zhuokapranov's,
  title={Kapranov's Construction of Homotopy Leibniz Algebras},
  author={Zhuo Chen, Zhangju Liu, and MAOSONG XIANG},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180907141245998366154},
}
Zhuo Chen, Zhangju Liu, and MAOSONG XIANG. Kapranov's Construction of Homotopy Leibniz Algebras. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180907141245998366154.
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