[论文解读] Homotopy Lie algebras and coherent infinitesimal 2-braidings

Cameron James Deverall Kemp|arXiv (Cornell University)|Feb 17, 2026

Homotopy and Cohomology in Algebraic Topology被引用 0

一句话总结

论文构建了 L∞-代数表示的对称单（monoidal） dg-谱类别，利用 2 位移 Poisson 结构推导 infinitesimal 2- braidings 并证明其一致性；并将表示与 Chevalley–Eilenberg 代数的半自由 dg-模相关联的派生几何视角联系起来。

ABSTRACT

Given a homotopy Lie algebra (i.e. an $L_\infty$-algebra) $\mathfrak{g}$, we show concretely how the Lada-Markl $\mathfrak{g}$-modules (i.e. representations) assemble into a symmetric monoidal dg-category. Considering the homotopy 2-category of that dg-category, we construct infinitesimal 2-braidings from 2-shifted Poisson structures then show that such infinitesimal 2-braidings are coherent in Cirio and Faria Martins' sense. We then explicitly determine the differential of the Chevalley-Eilenberg algebra associated with a finite-dimensional homotopy Lie algebra and construct the symmetric monoidal dg-equivalence between the category of representations and the category of semi-free dg-modules over the Chevalley-Eilenberg algebra.

研究动机与目标

• Motivate the study of L-infinity representations as structured dg-categories and seek a monoidal framework for higher representations.
• Define and construct a symmetric monoidal dg-category of g-representations via Lada–Markl data and dg-categorical enrichment.
• Introduce infinitesimal 2-braidings from 2-shifted Poisson structures and establish their coherence in Cirio–Faria Martins’ sense.
• Bridge representation theory with derived geometry by relating Rep(g) to semi-free dg-modules over Chevalley–Eilenberg algebras.

提出的方法

• Use the bullet/product formalism to encode n-shifted Poisson structures in a homotopy Lie algebra g.
• Explicitly define the symmetric monoidal dg-category Rep(g) and its intertwiners as a dg-category enriched structure.
• Construct infinitesimal 2-braidings from the weight-2 component of a 2-shifted Poisson structure and analyze coherence via Maurer–Cartan components.
• Demonstrate a symmetric monoidal dg-equivalence between Rep(g) and the category of semi-free dg-modules over the Chevalley–Eilenberg algebra of g.
• Employ a graphical calculus from prior work to realize representations and monoidal structure without relying on a CDGA Cartan formula.

实验结果

研究问题

• RQ1 How can representations of an L-infinity algebra be organized into a symmetric monoidal dg-category?
• RQ2 Can infinitesimal 2-braidings arising from 2-shifted Poisson structures be made coherent in the Cirio–Faria Martins sense?
• RQ3 What is the relationship between representation theory of g and derived geometry via Chevalley–Eilenberg algebras?
• RQ4 Under what conditions does a dg-equivalence with semi-free dg-modules exist between Rep(g) and CE(g)-modules?

主要发现

• A concrete symmetric monoidal dg-category Rep(g) for representations of a homotopy Lie algebra g is constructed.
• Infinitesimal 2-braidings t from 2-shifted Poisson structures are built and shown to be coherent under Cirio–Faria Martins’ framework.
• The differential of the Chevalley–Eilenberg algebra for a finite-dimensional g is determined explicitly.
• There is a symmetric monoidal dg-equivalence between Rep(g) and the category of semi-free dg-modules over CE(g).