∞-Lie theory

# Lie $2$-algebras

## Idea

A Lie 2-algebra is to a Lie 2-group as a Lie algebra is to a Lie group. Thus, it is a vertical categorification of a Lie algebra.

## Definition

### Semistrict case

A (“semistrict”) Lie 2-algebra $𝔤$ is an L-∞-algebra with generators concentrated in the lowest two degrees.

This means that it is

• a pair of vector spaces ${𝔤}_{0},{𝔤}_{1}$

• equipped with linear functions as follows:

a unary bracket $\left[-\right]$ encoding a differential

$\delta :{𝔤}_{1}\to {𝔤}_{0}\phantom{\rule{thinmathspace}{0ex}}$\delta : \mathfrak{g}_1 \to \mathfrak{g}_0 \,

and a binary bracket $\left[-,-\right]$, whose component on elements in degree 0 is a Lie bracket

$\left[-,-\right]:{𝔤}_{0}\vee {𝔤}_{0}\to {𝔤}_{0}$[-,-] : \mathfrak{g}_0 \vee \mathfrak{g}_0 \to \mathfrak{g}_0

and whose component on elements in degree 0 and degree 1 is a weak action

$\alpha \left(-,-\right):{𝔤}_{0}\otimes {𝔤}_{1}\to {𝔤}_{1}\phantom{\rule{thinmathspace}{0ex}};$\alpha(-,-) : \mathfrak{g}_0 \otimes \mathfrak{g}_1 \to \mathfrak{g}_1 \,;

and a trinary bracket

$\left[-,-,-\right]:{𝔤}_{0}\vee {𝔤}_{0}\vee {𝔤}_{0}\to {𝔤}_{1}$[-,-,-] : \mathfrak{g}_0 \vee \mathfrak{g}_0 \vee \mathfrak{g}_0 \to \mathfrak{g}_1

called the Jacobiator;

• such that

• $\left[-,-\right]$ and $\left[-,-,-\right]$ are skew-symmetric in their arguments, as indicated;

• the differential respects the brackets: for all $x\in {𝔤}_{0}$ and $h\in {𝔤}_{1}$ we have

$\delta \left[x,h\right]=\left[x,\delta h\right]$\delta [x,h] = [x, \delta h]

hence

$\delta \alpha \left(x,h\right)=\left[x,\delta h\right]\phantom{\rule{thinmathspace}{0ex}};$\delta \alpha(x,h) = [x, \delta h] \,;
• the Jacobi identity of $\left[-,-\right]$ holds up to the image under $\delta$ of the Jacobiator $\left[-,-,-\right]$: for all $x,y,z\in {𝔤}_{0}$ we have

$\left[x,\left[y,z\right]\right]+\left[y,\left[z,x\right]\right]+\left[z,\left[x,y\right]\right]=\delta \left[x,y,z\right]$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = \delta [x,y,z]
• as does the action property:

$\alpha \left(x,\left[y,h\right]\right)-\alpha \left(y,\left[x,h\right]\right)=\alpha \left(\left[x,y\right],h\right)+\left[x,y,\delta h\right]$\alpha(x,[y,h]) - \alpha(y,[x,h]) = \alpha([x,y],h) + [x,y,\delta h]
• the Jacobiator is coherent:

$\left[\left[w,x,y\right],z\right]+\left[\left[w,y,z\right],x\right]+\left[\left[w,y\right],x,z\right]+\left[\left[x,z\right],w,y\right]=\left[\left[w,x,z\right],y\right]+\left[\left[x,y,z\right],w\right]+\left[\left[w,x\right],y,z\right]+\left[\left[w,z\right],x,y\right]+\left[\left[x,y\right],w,z\right]+\left[\left[y,z\right],w,x\right]\phantom{\rule{thinmathspace}{0ex}}.$[[w,x,y], z] + [[w,y,z],x] + [[w,y],x,z] + [[x,z],w,y] = [[w,x,z], y] + [[x,y,z], w] + [[w,x],y,z] + [[w,z], x,y] + [[x,y], w,z] + [[y,z],w,x] \,.

### Strict case

If the trinary bracket $\left[-,-,-\right]$ in a Lie 2-algebra is trivial, one speaks of a strict Lie 2-algebra. Strict Lie 2-algebras are equivalently differential crossed modules (see there for details).

## References

Revised on February 28, 2012 19:39:59 by Urs Schreiber (131.174.40.190)