CLWX 2-代数胚、左对称代数胚与Poisson几何

CLWX 2-代数胚是4维拓扑场论关于AKSZ构造中靶空间所对应的几何结构，与Courant代数胚的范畴化和二次李2-代数胚的第一Pontryagin类有密切联系。左对称代数胚是左对称代数的几何化，其交换子是一个李代数胚，这使得它在Hessian几何、仿射几何和几何力学等领域中有着重要的应用。.首先，我们研究CLWX 2-代数胚的Dirac结构，并将其应用到李2-代数胚的Manin三元组和同伦Poisson代数的Maurer-Cartan元的理论中。然后，我们研究Koszul-Vinberg流形的局部结构，在Koszul-Vinberg流形上建立类似于Poisson流形上的Weinstein可裂定理。最后，我们研究Koszul-Vinberg-Nijenhuis结构和pseudo-Hessian-Nijenhuis结构的几何性质以及它们之间的关系。

CLWX 2-algeboid is the geometrical structure of the target space of AKSZ(Alexandrov-Kontsevich-Schwarz-Zaboronsky) construction in 4-dimensional topological field theory, which has a close relationship with the categorification of the Courant algebroid and the first Pontryagin class of the quadratic Lie 2-algebroid. The notion of a left-symmetric algebroid is a geometric generalization of a left-symmetric algebra, whose commutator gives rise to a Lie algebroid such that it has an important application in Hessian geometry, affine geometry, geometric mechanics, etc. .First，we study the Dirac structures of CLWX 2-algebroids, and apply these results to the theories of Manin triple for Lie 2-algebroids and Maurer-Cartan elements of the homotopy Poisson algebras. Then we study the local structure of Koszul-Vinberg manifolds, and build an analogue of the Weinstein splitting theorem of the Poisson manifolds on Koszul-Vinberg manifolds. At last, we study the geometric properties and relations of Koszul-Vinberg-Nijenhuis structures and pseudo-Hessian-Nijenhuis structures.