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 \(X:G\rightarrow \mathfrak {g}\). 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 \(\alpha :V\rightsquigarrow \mathfrak {g}\). 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 also 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 \(\hbox {Leibniz}_\infty [1]\) 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.