Semidirect products of representations up to homotopy

Abstract

We study the semidirect product of a Lie algebra with a representation up to homotopy and provide various examples coming from Courant algebroids, Lie 2-algebras of string type, and omni-Lie algebroids.In the end, we study the semidirect product of a Lie group with a representation up to homotopy and use it to give an integration of a certain Lie 2-algebra of string type.