# nLab Lie operad

∞-Lie theory

## Examples

### $\infty$-Lie algebras

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

Given an ambient additive symmetric monoidal category, the Lie operad is the operad whose algebras over an operad are Lie algebras in that category. It is a quadratic operad? whose Koszul dual is the operad for commutative algebras.

In the case of the monoidal category of chain complexes, we also say dg-Lie-operad. Its cofibrant resolution in the model structure on operads is the L-infinity operad whose algebras over an operad are L-∞-algebras.

## References

• V. Hinich, Homological algebra of homotopy algebras, Communications in algebra, 25(10) (1997), 3291-3323, pdf; erratum math.AG/0309453

Revised on August 13, 2012 10:37:52 by Beren Sanders (96.251.14.54)