Dg Loday-Pirashvili modules over Lie algebras

Abstract

A Loday-Pirashvili module over a Lie algebra $\mathfrak{g}$ is a Lie algebra object $\bigl(G\xrightarrow{X} \mathfrak{g} \bigr)$ in the category of linear maps, or equivalently, a $\mathfrak{g}$-module $G$ which admits a $\mathfrak{g}$-equivariant linear map $X:G\to \mathfrak{g}$. We study dg Loday-Pirashvili modules over Lie algebras, which is a generalization of Loday-Pirashvili modules in a natural way, and establish several equivalent characterizations of dg Loday-Pirashvili modules. To provide a concise characterization, a dg Loday-Pirashvili module is a non-negative and bounded dg $\mathfrak{g}$-module $V$ paired with a weak morphism of dg $\mathfrak{g}$-modules $\alpha\colon V\rightsquigarrow \mathfrak{g}$. Such a dg Loday-Pirashvili module resolves an arbitrarily specified classical Loday-Pirashvili module in the sense that it exists and is unique (up to homotopy). Dg Loday-Pirashvili modules can be characterized through dg derivations. This perspective allows the calculation of the corresponding twisted Atiyah classes. By leveraging the Kapranov functor on the dg derivation arising from a dg Loday-Pirashvili module $(V,\alpha)$, a Leibniz$_\infty[1]$ algebra structure can be derived on $\wedge^\bullet \mathfrak{g}^\vee\otimes V[1]$. The binary bracket of this structure corresponds to the twisted Atiyah cocycle. To exemplify these intricate algebraic structures through specific cases, we utilize this machinery to a particular type of dg Loday-Pirashvili modules stemming from Lie algebra pairs.