Semidirect products of representations up to homotopy

TL;DR

This paper explores the construction of semidirect products involving Lie algebras and Lie groups with representations up to homotopy, providing new examples and an integration approach for string Lie 2-algebras.

Contribution

It introduces the concept of semidirect products with representations up to homotopy and applies it to various structures like Courant algebroids and string Lie 2-algebras.

Findings

Examples from Courant algebroids, string Lie 2-algebras, and omni-Lie algebroids.

A method for integrating certain string Lie 2-algebras.

New insights into the structure of semidirect products in higher Lie theory.

Abstract

We study the semidirect product of a Lie algebra with a representation up to homotopy and provide various examples coming from Courant algebroids, string Lie 2-algebras, 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 string Lie 2-algebra.