Abstract
We give an overview of different characterisations of L∞-structures in terms of symmetric brackets and (co)differentials on the symmetric (co)algebra. We then do the same for their representations (up to homotopy) and approach L∞-algebra cohomology using the commutator bracket on the space of coderivations of the symmetric coalgebra. This leads to abelian extensions of L∞-algebras by 2-cocycles.
Highlights
L∞-algebras were first introduced in [1] and [2] and are a generalisation of graded Lie algebras in which a system of antisymmetric n-ary brackets satisfies a generalised Jacobi identity
The first part of this article serves as a self-contained introduction to L∞-algebras, in which we discuss different characterisations of L∞-algebras and their representations, closely following [3]
The L∞-algebra cohomology with values in the adjoint representation was introduced in [4] using a Lie bracket on the space of cochains
Summary
L∞-algebras ( called strongly homotopy Lie algebras) were first introduced in [1] and [2] and are a generalisation of graded Lie algebras in which a system of antisymmetric n-ary brackets satisfies a generalised Jacobi identity. The first part of this article serves as a self-contained introduction to L∞-algebras, in which we discuss different characterisations of L∞-algebras and their representations (up to homotopy), closely following [3]. The L∞-algebra cohomology with values in the adjoint representation was introduced in [4] using a Lie bracket on the space of cochains We extend this approach to arbitrary representations, which leads to a characterisation of certain L∞-algebras as abelian extensions of L∞-algebras by 2-cocycles. This generalises a theorem from [5] that characterises certain L∞-algebras in terms of Lie algebra cohomology. This article is largely based on my same-titled Bachelor’s thesis, which I wrote under the supervision of Chenchang Zhu at the University of Gottingen in 2018
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