发表机构
Université Bourgogne Europe, CNRS, IMB (UMR 5584); Tokyo Denki University（勃艮第欧洲大学，法国国家科学研究中心，数学研究所; 东京电机大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究同调圆柱李代数挠元的Sp-模结构，通过改进LMO函子模1约化与钳子演算工具，证明挠部分满射到2-挠Sp-模，并给出三次分量的内在描述，进而理解Torelli群挠现象。
AI 中文摘要
设 $\Sigma$ 为一个紧致可定向曲面。我们研究由 M. Goussarov 和 K. Habiro 引入的 $\Sigma$ 上同调圆柱的分次李代数。为分析其挠元，我们改进了由 Y. Nozaki、M. Suzuki 和第三作者发起的一种策略，该策略依赖于 LMO 函子的模 $1$ 约化以及钳子演算。具体而言，我们发展了通用工具，用于在辛群的标准作用下研究该李代数奇次分量中挠元的 Sp-模结构。我们证明该挠部分满射到一个 $2$-挠 Sp-模，该模用雅可比图显式描述。作为应用，我们给出了由同调圆柱李代数三次分量给出的 Sp-模的内在描述。进一步的动机是理解 $\Sigma$ 的 Torelli 群的下中心序列的伴随分次中的挠现象：在这一方向上，我们展示了一个显式的 Sp-模，Torelli 李代数在三次中的挠元满射到该模上。
英文摘要
Let $Σ$ be a compact oriented surface. We investigate the graded Lie algebra of homology cylinders over $Σ$, as introduced by M. Goussarov and K. Habiro. To analyze its torsion, we refine a strategy initiated by Y. Nozaki, M. Suzuki, and the third author, which relies on the reduction modulo $1$ of the LMO functor and on clasper calculus. Specifically, we develop general tools for studying, under the standard action of the symplectic group, the Sp-module structure of the torsion in the odd-degree component of this Lie algebra. We show that this torsion part surjects onto an $2$-torsion Sp-module, which is explicitly described in terms of Jacobi diagrams. As an application, we provide an intrinsic description of the Sp-module given by the degree-three component of the Lie algebra of homology cylinders. A further motivation is to understand torsion phenomena in the associated graded of the lower central series of the Torelli group of $Σ$: in this direction, we exhibit an explicit Sp-module onto which the torsion of the Torelli Lie algebra surjects in degree three.
Comments76 pages
