---
created: 2022-02-13T20:29
updated: 2023-04-12T12:48
title: L infty algebra
aliases: ["L infty algebra", "L_infty algebra", "L_infty", "L infty", DGLA]
---

---

- Tags 
	- #todo/untagged
- Refs:
	- #todo/add-references
- Links:
	- [A_infty](Unsorted/A_infty.md)
	- [Maurer-Cartan](Maurer-Cartan.md)
	- [Chevalley-Eilenberg complex](Unsorted/Chevalley-Eilenberg%20complex.md)
	- [BRST](BRST)
	- [Lie algeberoid](Lie%20algeberoid.md)
	- [BV formalism](Unsorted/BV%20algebra.md)
	- [DGA](DGA.md)

---



# L infty algebra

Generalizes [A_infty](A_infty.md) algebras from multiplications to brackets. 
DGLAs are a special case of $L_\infty$ algebras.

![](2023-03-31-64.png)

![](2023-03-31-71.png)

[Maurer-Cartan](Maurer-Cartan.md) element and relation to DGLAs: ![](2023-03-31-72.png)

![](attachments/2023-03-31.png)

As an alternative to $Q\dash$manifolds:

![](attachments/2023-03-31-1.png)
![](attachments/2023-03-31-2.png)

- The role of zero locus of $Q$ is played by the space of solutions of Maurer-Cartan (MC) equation:
$$
\sum_{n} \frac{1}{n !} \mu_{n}(a, \ldots, a)=0 .
$$
	- For [A_infty algebras](Unsorted/A_infty%20algebra.md), the role of equations of motion is played by the MC equation
$$
\sum_{n \geq 1} \mu_{n}(a, \ldots, a)=0
$$

- An $L_{\infty}$ algebra specifies a classical system and $L_{\infty}$ algebra with invariant odd inner product specifies a Lagrangian classical system.

-  A $\mathbb{Z}$-graded $L_{\infty}$-algebra in [BV formalism](BV%20formalism) corresponds to the case when the fields are classified according to the ghost number.
- An $\mathrm{L}_{\infty}$ algebra where all operations $\mu_{n}$ with $n \geq 3$ vanish can be identified with differential graded Lie algebra (the operation $\mu_{1}$ is the differential, $\mu_{2}$ is the bracket). 
- An $\mathrm{L}_{\infty}$ algebra corresponding to a dg Lie algebra with zero differential is $\mathbb{Z}$-graded.
- Considering gauge theories for all $\U_n$ at the same time: it is more convenient to work with $\mathrm{A}_{\infty}$ instead of $\mathrm{L}_{\infty}$ algebras. See [A_infty](Unsorted/A_infty.md).

# Derived brackets

![](attachments/2023-03-31-3.png)
![](attachments/2023-03-31-4.png)
![](attachments/2023-03-31-5.png)