# nLab Jacobi identity

This entry is about the identity in Lie theory. For another identity named after Jacobi, see at Jacobi theta-function and Jacobi form.

∞-Lie theory

# Contents

## Idea

The Jacobi identity is an important equational identity that holds in Lie algebras and is also of interest in other algebras. It can be generalised to higher algebras.

## For $1$-algebras

### Definition

Given a nonassociative algebra $\mathfrak{g}$ over a field or a ring $k$, whose bilinear product is denoted by bracket $[,]:\mathfrak{g}\otimes\mathfrak{g}\to\mathfrak{g}$, the Jacobi identity holds for a triple of elements $x,y,z \in \mathfrak{g}$ if

$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 \,.$

The principal example is that of a Lie algebra: here the Jacobi identity by definition holds for all triples of elements (and the bracket is skew-symmetric).

### Relation to the Leibniz identity

If the bracket $[-,-]$ is skew-symmetric the Jacobi identity for all triples is equivalent to the Leibniz identity that for all $x,a,b$, the linear map $ad_x = [x,-] : \mathfrak{g} \to \mathfrak{g}$ is a derivation with respect to the Lie bracket:

$[x,[a,b]] = [[x,a],b] + [a,[x,b]] \,.$

One can also consider right derivations (right Leibniz identity), what is again equivalent in the presence of antisymmetry:

$[[a,b],x] = [[a,x],b] + [a,[b,x]] \,.$

Left and right Leibniz algebras generalize Lie algebras by having a left or right Leibniz identity, but not necessarily antisymmetry of the bracket. In particular, Jacobi identity, and left and right Leibniz identities do not need to coincide without antisymmetry.

### In terms of Chevalley–Eilenberg algebras

A useful way to think of Jacobi identity for finite-dimensional Lie algebras, is dually in terms of the Chevalley–Eilenberg algebra $CE(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ (see there for details). In terms of this dg-algebra $(\wedge^\bullet \mathfrak{g}^*, d_{\mathfrak{g}})$, the Lie bracket is encoded in the differential

$d_{\mathfrak{g}}|_{\mathfrak{g}^*} := [-,-]^* : \mathfrak{g}^* \to \mathfrak{g}^* \wedge \mathfrak{g}^*$

and the Jacobi identity is equivalent to the statement that this differential squares to $0$

$d \circ d = 0 \,.$

## For $L_\infty$-algebras

When Lie algebras are generalized to ∞-Lie algebras, the Jacobi identity in terms of the binary bracket is relaxed to hold only up to a natural isomorphism called the jacobiator, $[-,-,-] : \mathfrak{g}_0 \vee \mathfrak{g}_0 \vee \mathfrak{g}_0 \to \mathfrak{g}_1$,

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

where $\delta$ is the differential.

On the other hand, in terms of the Chevalley–Eilenberg algebra this is still encoded in just $d \circ d = 0$ (see there for details).

Revised on May 17, 2014 20:10:08 by Tim Porter (2.26.27.252)