Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dg Loday-Pirashvili modules over Lie algebras   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 $X:G\to \mathfrak{g}$. In this note, we
introduce the notion of 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. In short,
a dg Loday-Pirashvili module is a non-negative and bounded dg
$\mathfrak{g}$-module $V$ together with a weak morphism of dg
$\mathfrak{g}$-modules $\alpha\colon V\rightsquigarrow \mathfrak{g}$. Such dg
Loday-Pirashvili modules can be characterized through dg derivations, which in
turn allows us to calculate the associated twisted Atiyah classes. By applying
the Kapranov functor to the dg derivation arising from a dg Loday-Pirashvili
module $(V,\alpha)$, we obtain a Leibniz$_\infty[1]$ algebra structure on
$\wedge^\bullet \mathfrak{g}^\vee\otimes V[1]$ whose binary bracket is the
twisted Atiyah cocycle. Finally, we apply this machinery to a natural type of
dg Loday-Pirashvili modules stemming from Lie algebra pairs.

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The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: nlin
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The categories of this paper are: hep-lat
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The categories of this paper are: dg-ga
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The categories of this paper are: chem-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: eess
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The categories of this paper are: nucl-th
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: adap-org