nLab
Lie 2-algebra

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 $[-]$ encoding a differential

  $$\delta :{𝔤}_1\to {𝔤}_0$$\delta : \mathfrak{g}_1 \to \mathfrak{g}_0 \,

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

  $$[-,-]:{𝔤}_0\vee {𝔤}_0\to {𝔤}_0$$[-,-] : \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

  $$\alpha (-,-):{𝔤}_0\otimes {𝔤}_1\to {𝔤}_1;$$\alpha(-,-) : \mathfrak{g}_0 \otimes \mathfrak{g}_1 \to \mathfrak{g}_1 \,;

  and a trinary bracket

  $$[-,-,-]:{𝔤}_0\vee {𝔤}_0\vee {𝔤}_0\to {𝔤}_1$$[-,-,-] : \mathfrak{g}_0 \vee \mathfrak{g}_0 \vee \mathfrak{g}_0 \to \mathfrak{g}_1

  called the Jacobiator;

• such that

  • $[-,-]$ and $[-,-,-]$ 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

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

    hence

    $$\delta \alpha (x,h)=[x,\delta h];$$\delta \alpha(x,h) = [x, \delta h] \,;

  • the Jacobi identity of $[-,-]$ holds up to the image under $\delta$ of the Jacobiator $[-,-,-]$: for all $x,y,z\in {𝔤}_0$ we have

    $$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=\delta [x,y,z]$$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = \delta [x,y,z]

  • as does the action property:

    $$\alpha (x,[y,h])-\alpha (y,[x,h])=\alpha ([x,y],h)+[x,y,\delta h]$$\alpha(x,[y,h]) - \alpha(y,[x,h]) = \alpha([x,y],h) + [x,y,\delta h]

  • the Jacobiator is coherent:

    $$[[w,x,y],z]+[[w,y,z],x]+[[w,y],x,z]+[[x,z],w,y]=[[w,x,z],y]+[[x,y,z],w]+[[w,x],y,z]+[[w,z],x,y]+[[x,y],w,z]+[[y,z],w,x].$$[[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 $[-,-,-]$ 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).

Examples

• derivation Lie 2-algebra

• string Lie 2-algebra

Related concepts

• Lie algebra, Lie group

• Lie 2-algebra, Lie 2-group

• L-∞ algebra, smooth ∞-group

• Lie algebroid, Lie groupoid

• L-∞ algebroid, smooth ∞-groupoid

References

• John Baez, Alissa Crans, Higher-Dimensional Algebra VI: Lie 2-Algebras Theory and Applications of Categories, Vol. 12, (2004) No. 15, pp 492-528. (TAC)

• Dmitry Roytenberg, On weak Lie 2-algebras, (arXiv)