Higher holonomy for curved L${}_\infty$-algebras 1: simplicial methods

Ezra GetzlerNorthwestern University
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引用次数: 0

Abstract

We construct a natural morphism $\rho$ from the nerve $\text{MC}_\bullet(L) = \text{MC}(\Omega_\bullet \widehat{\otimes} L)$ of a pronilpotent curved L${}_\infty$-algebra $L$ to the simplicial subset $\gamma_\bullet(L) = \text{MC}(\Omega_\bullet \widehat{\otimes} L,s_\bullet)$ of Maurer--Cartan element satisfying the Dupont gauge condition. This morphism equals the identity on the image of the inclusion $\gamma_\bullet(L) \hookrightarrow \text{MC}_\bullet(L)$. The proof uses the extension of Berglund's homotopical perturbation theory for L${}_\infty$-algebras to curved L${}_\infty$-algebras. The morphism $\rho$ equals the holonomy for nilpotent Lie algebras. In a sequel to this paper, we use a cubical analogue $\rho^\square$ of $\rho$ to identify $\rho$ with higher holonomy for semiabelian curved \Linf-algebras.
曲线 L${}_\infty$-algebras 的高整体性 1:简约方法
我们构建了一个自然态量 $\rho$,它从一个代potent curvedL${}_\infty$-algebra $L$ 的神经 $\text{MC}_\bullet(L) =\text{MC}(\Omega_\bullet \widehat\{otimes} L)$ 到简单子集 $\gamma_\bullet(L) =\text{MC}(\Omega_\bullet \widehat\{otimes} L. s_\bullet)$、s_\bullet)$ 的毛勒卡尔元素满足杜邦轨距条件。这个变形等价于包含 $\gamma_\bullet(L)\hookrightarrowtext{MC}_\bullet(L)$ 的图像上的同一性。证明使用了贝格伦德关于 L${}_infty$-algebras 的同域扰动理论对弯曲 L${}_infty$-algebras 的扩展。在本文的续篇中,我们使用$\rho$的立方类似物$\rho^\square$来识别$\rho$与半阿贝尔弯曲\Linf-gebras的高整体性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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