Operadic deformations as a tool for cogravity

An operadic approach to deformation quantization of compatible Poisson brackets

We construct an analogue of the Livernet--Loday operad for two compatible brackets. The Livernet--Loday operad can be used to define $\star$-products and deformation quantization for Poisson structures. We make use of our operad in the same way, introducing a definition of (operadic) deformation quantization of compatible Poisson structures.

On higher structure on the operadic deformation complexes ${Def}(e_n\to \mathcal{P})$

In this paper, we prove that there is a canonical homotopy $(n+1)$-algebra structure on the shifted operadic deformation complex $Def(e_n\to\mathcal{P})[-n]$ for any operad $\mathcal{P}$ and a map of operads $f\colon e_n\to\mathcal{P}$. This result generalizes the result of [T2], where the case $\mathcal{P}=\mathrm{End}_{Op}(X)$ was considered. Another more computational proof of the same statement was recently sketched in [CW]. Our method combines the one of [T2] with the categorical algebra on the category of symmetric sequences, introduced in [R] and further developed in [KM] and [Fr1]. We …

Operad 形变理论

Operadic Deformation Theory

本文综述代数 operad 与 Koszul 对偶等同伦工具在代数形变理论中的应用，并从 operad 视角考察形式模问题。

代数几何与变形量子化中的导出运算中心

Derived operadic centers in algebraic geometry and deformation quantization

研究代数几何中霍赫希尔德上同调相关问题，通过将霍赫希尔德复形展示为\(\infty\)-运算中心，给出德利涅猜想典范解，为其配备通用\(\mathbb{E}_2\)-代数结构，并从运算形式模问题角度解释该结构。

A concept of $\frac23$PROP and deformation theory of (co)associative bialgebras

We introduce a concept of $\frac23$PROP generalizing the Kontsevich concept of $\frac12$PROP. We prove that some Stasheff-type compactification of the Kontsevich spaces $K(m,n)$ defines a topological $\frac23$PROP structure. The corresponding chain complex is a minimal model for its cohomology (both are considered as $\frac23$PROPs). We construct a $\frac23$PROP $\End(V)$ for a vector space $V$. Finally, we construct a dg Lie algebra controlling the deformations of a (co)associative bialgebra. Philosophically, this construction is a version of the Markl's operadic construction from [M1] applie…