论文标题
局部有限的无限模型和弱loday-pirashvili模块在差分级代数上
Locally finite infinity-modules and weak Loday-Pirashvili modules over differential graded Lie algebras
论文作者
论文摘要
在最近的$ \ infty $ - 类别理论的最新发展与差分（简称DG）相关的代数方面，我们为本地有限的$ \ infty $ - $ \ mathfrak {g} $ - 模块模块开发了一个通用框架。我们表明，这种本地有限的$ \ infty $ - $ \ mathfrak {g} $ - 模块几乎是Vallette意义上的模型类别。作为线性图的张量类别中的Loday和Pirashvili的Lie代数对物的同源理论概括，我们进一步研究了由$ \ infty $ - morphisms组成的Loday-Pirashvili模块，该模块由本地有限的$ \ infty $ \ infty $ \ infty $ - $ \ $ \ $ \ $ \ $ \ $ \ $ \ $ \ $ \ $ \ $ \ $ \ $ \ $ \ $ \ $ \ \ $ \ $ \ $ \ \ \ \ \ \ $ \ $ \ $ \ \ $ \ $ \ \ \ \ - $ \ mathfrak {g} $。从$ \ mathfrak {g} $上方的如此弱的loday-pirashvili模块的类别中，我们找到了一个函数，将其映射到Leibniz $ _ \ Infty $ algebras的类别，该类别富含Chevalley-Eilenberg dg algebra of Chevalley-eilenberg dg algebra of $ \ m m ialfrak $ \ m m mathfrak {g} $ {g} $。该函子可以被视为Loday和Pirashvili的原始方法的同质抬起，以实现线性图类别中Lie代数对象的Leibniz代数。
Motivated by recent developments of $\infty$-categorical theories related to differential graded (dg for short) Lie algebras, we develop a general framework for locally finite $\infty$-$\mathfrak{g}$-modules over a dg Lie algebra $\mathfrak{g}$. We show that the category of such locally finite $\infty$-$\mathfrak{g}$-modules is almost a model category in the sense of Vallette. As a homotopy theoretical generalization of Loday and Pirashvili's Lie algebra objects in the tensor category of linear maps, we further study weak Loday-Pirashvili modules consisting of $\infty$-morphisms from locally finite $\infty$-$\mathfrak{g}$-modules to the adjoint module $\mathfrak{g}$. From the category of such weak Loday-Pirashvili modules over $\mathfrak{g}$, we find a functor that maps to the category of Leibniz$_\infty$ algebras enriched over the Chevalley-Eilenberg dg algebra of $\mathfrak{g}$. This functor can be regarded as the homotopy lifting of Loday and Pirashvili's original method to realize Leibniz algebras from Lie algebra objects in the category of linear maps.
