Character formulas for the operad of two compatible brackets and for the bihamiltonian operad

源自相对移位 Poisson 结构的椭圆双 Hamilton 结构

Elliptic bihamiltonian structures from relative shifted Poisson structures

该研究推广相对移位 Poisson 结构构造，用于 Calabi–Yau 纤维化上复形的相对模叠，并由 Hirzebruch 曲面显式恢复并推广 Feigin–Odesskii 及 Odesskii–Wolf 椭圆 Poisson 括号。

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.

Derived bracket construction up to homotopy and Schroder numbers

We will introduce the notion of higher derived bracket construction in the category of operads and prove that the higher derived bracket construction of Lie operad is equivalent to the cobar construction of Leibniz operad. The theorem is proved by computing Schroder number.

Derived bracket construction and Manin products

We will extend the classical derived bracket construction to any algebra over a binary quadratic operad. We will show that the derived product construction is a functor given by the Manin white product with the operad of permutation algebras. As an application, we will show that the operad of prePoisson algebras is isomorphic to Manin black product of the Poisson operad with the preLie operad. We will show that differential operators and Rota-Baxter operators are, in a sense, Koszul dual to each other.

共享一个李括号的 pre-Lie 乘积的组合学

Combinatorics of pre-Lie products sharing a Lie bracket

研究共享同一李括号的多个 pre-Lie 代数结构所对应的 operad，证明其具有类似 pre-Lie operad 的组合描述及诸多代数性质。

双结合（diassociative）与树状（dendriform）operad的单参数形变

One-parameter deformations of the diassociative and dendriform operads

本研究结合Livernet和Loday提出的结合operad极化构造与结合运算的树状分裂方法，借助二次operad的Koszul对偶，实现了非对称树状、双结合operad到对称operad范畴的单参数形变。