发表机构
Sobolev Institute of Mathematics; Jilin University（西伯利亚科学院数学研究所; 吉林大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过 operad 方法刻画 Novikov 代数上同调，给出显式链复形，并证明其与底层 pre-Lie 代数上同调存在长正合序列，且第二上同调群分类无穷小形变与 Abel 扩张。
AI 中文摘要
本文首先利用与从 Lie 代数 operad 到 pre-Lie 代数 operad 与其 Koszul 对偶 operad 的 Hadamard 积的态射相关联的 Chevalley-Eilenberg 上同调，给出了 pre-Lie 代数上同调的一个新刻画。然后我们将同样的方法应用于研究 Novikov 代数的上同调，并显式地给出了链复形。我们证明了底层 pre-Lie 代数的链复形同构于 Novikov 代数链复形的商。因此，存在一个长正合序列连接 Novikov 代数与其底层 pre-Lie 代数的上同调。利用伪张量范畴，引入了系数在表示中的 Novikov 代数上同调。作为应用，我们证明了无穷小形变和 Abel 扩张由不同系数的第二上同调群分类。我们给出了各种例子来说明 Novikov 代数上同调与其底层 pre-Lie 代数上同调之间的差异。
英文摘要
In this paper, first we give a new characterization of the cohomology of pre-Lie algebras using the Chevalley-Eilenberg cohomology associated to a morphism from the operad of Lie algebras to Hadamard product of the operad of pre-Lie algebras and its Koszul dual operad. Then we apply the same approach to study the cohomology of Novikov algebras, and give the cochain complex explicitly. The cochain complex of the underlying pre-Lie algebra is shown to be isomorphic to the quotient of the cochain complex of a Novikov algebra. Consequently, there is a long exact sequence connecting the cohomologies of a Novikov algebra and the underlying pre-Lie algebra. The cohomology of a Novikov algebra with coefficients in a representation is introduced using pseudo-tensor categories. As applications, we show that infinitesimal deformations and abelian extensions are classified by the second cohomology groups with different coefficients. Various examples are given to illustrate the difference between the cohomology of a Novikov algebra and that of the underlying pre-Lie algebra.
Comments36 pages
