nLab
Lie 2-algebra

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Lie 22-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\mathfrak{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\mathfrak{g}_0, \mathfrak{g}_1

  • equipped with linear functions as follows:

    a unary bracket [][-] encoding a differential

    δ:𝔤 1𝔤 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 0𝔤 0𝔤 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

    α(,):𝔤 0𝔤 1𝔤 1; \alpha(-,-) : \mathfrak{g}_0 \otimes \mathfrak{g}_1 \to \mathfrak{g}_1 \,;

    and a trinary bracket

    [,,]:𝔤 0𝔤 0𝔤 0𝔤 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 x𝔤 0x \in \mathfrak{g}_0 and h𝔤 1h \in \mathfrak{g}_1 we have

      δ[x,h]=[x,δh] \delta [x,h] = [x, \delta h]

      hence

      δα(x,h)=[x,δh]; \delta \alpha(x,h) = [x, \delta h] \,;
    • the Jacobi identity of [,][-,-] holds up to the image under δ\delta of the Jacobiator [,,][-,-,-]: for all x,y,z𝔤 0x,y,z \in \mathfrak{g}_0 we have

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

      α(x,[y,h])α(y,[x,h])=α([x,y],h)+[x,y,δh] \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

References

Revised on December 11, 2013 07:26:21 by Urs Schreiber (89.204.130.40)