---
created: 2022-02-07T23:01
updated: 2023-04-02T14:08
title: DGA
aliases: [DGA, DGAs, DGLAs, differential graded algebra, differential graded algebras, CDGA, CDGAs]
---

---

- Tags 
	- #AG/deformation-theory 
- Refs:
	- #todo/add-references
- Links:
	- [bar construction](Unsorted/bar%20construction.md)
	- [Poisson structure](Unsorted/Poisson%20structure.md)
	- [graded mixed algebra](graded%20mixed%20algebra.md)
	- [Maurer-Cartan](Maurer-Cartan.md)
	- [additive category](additive%20category)
	- [DGLA](L%20infty%20algebra.md)

---

# DGAs

GAs: ![](2023-04-02-22.png)

DGAs:

![](2023-03-31-11.png)
![](2023-03-31-12.png)


![](attachments/Pasted%20image%2020220505193935.png)

![](attachments/Pasted%20image%2020220207230137.png)
![](attachments/Pasted%20image%2020220207230147.png)

![](attachments/Pasted%20image%2020220505194057.png)

## Hopf algebras

![](attachments/Pasted%20image%2020220207230222.png)
![](attachments/Pasted%20image%2020220207230257.png)

![](attachments/Pasted%20image%2020220207230353.png)
![](attachments/Pasted%20image%2020220207230411.png)

Opposite and enveloping algebras:
![](attachments/Pasted%20image%2020220207230527.png)

## Properties

![](attachments/Pasted%20image%2020220505193920.png)

## Misc

![](attachments/Pasted%20image%2020220317200542.png)

## DG Lie Algebras

![](attachments/Pasted%20image%2020220207230643.png)

A Gerstenhaber algebra is a graded $k$-module $A$ together with a graded-commutative
multiplication and a degree-1 Lie bracket that are compatible via the Poisson relation
$$
[a, b c]=[a, b] c+(-1)^{|b|(|a|-1)} b[a, c] .
$$
on homogeneous elements $a, b, c \in A$,

See [BV algebra](BV%20algebra.md)

See [A_infty](Unsorted/A_infty.md):
![](attachments/Pasted%20image%2020220207231256.png)
![](attachments/Pasted%20image%2020220207231317.png)

# Definitions

![](attachments/Pasted%20image%2020220213224339.png)
![](attachments/Pasted%20image%2020220213224440.png)
![](attachments/Pasted%20image%2020220213224518.png)
![](attachments/Pasted%20image%2020220213224601.png)
![](attachments/Pasted%20image%2020220213224630.png)

## Minimal models for spheres
![](attachments/Pasted%20image%2020220213224819.png)

## Formal DGAs
![](attachments/Pasted%20image%2020220213224849.png)
## Homotopy Groups of a DGA

![](attachments/Pasted%20image%2020220213224723.png)

# Formality

![](attachments/Pasted%20image%2020220323183314.png)


# Spheres
![](attachments/Pasted%20image%2020220505194119.png)

Relation to [A_infty algebras](A_infty.md): ![](2023-03-31-114.png)