形变理论综述(一):代数结构形变的具体公式

Review of Deformation Theory Ⅰ:Concrete Formulas for Deformations of Algebraic Structures

在这篇综述中,我们首先给出结合代数、李代数、预李代数、Leibniz代数以及3-李代数的表示和上同调的具体公式以及它们的强同伦版本.然后我们回顾可以刻画这些代数结构为Maurer-Cartan元的分次李代数和分次结合代数.相应的Maurer-Cartan元可以赋予分次李代数或者分次结合代数一个新的微分,进而给定代数结构的形变问题可以由得到的微分分次李代数或者微分分次结合代数的Maurer-Cartan元来刻画.我们还回顾了控制形变的上同调、微分分次李代数和微分分次结合代数之间的关系.

In this review,first we give the concrete formulas of representations and cohomologies of associative algebras,Lie algebras,pre-Lie algebras,Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues.Then we recall the graded Lie algebras and graded associative algebras that characterize these algebraic structures as Maurer-Cartan elements.The corresponding Maurer-Cartan element equips the graded Lie or associative algebra with a differential.Then the deformations of the given algebraic structures are characterized as the Maurer-Cartan elements of the resulting differential graded Lie or associative algebras.We also recall the relation between the cohomologies and the differential graded Lie and associative algebras that control the deformations.