Jacobi identity interpretations

The Jacobi identity in a Lie algebra $\mathfrak{g}$ has various interpretations that are more transparent, whence easier to remember, than the usual form

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

One is the fact that the adjoint representation ¹¹Here, “ $\mathfrak{gl}(\mathfrak{g})$ ” means the space o endomorphisms of $\mathfrak{g}$ , viewed as a vector space, with Lie bracket on $\mathfrak{gl}(\mathfrak{g})$ being commutator. $\operatorname{ad}:\mathfrak{g}\rightarrow\mathfrak{gl}(\mathfrak{g})$ really is a representation. Yet another way to formulate the identity is

$\operatorname{ad}(x)[y,z]=[\operatorname{ad}(x)y,z]+[y,\operatorname{ad}(x)z],$

i.e., $\operatorname{ad}(x)$ is a derivation on $\mathfrak{g}$ for all $x\in\mathfrak{g}$ .

Title	Jacobi identity interpretations
Canonical name	JacobiIdentityInterpretations
Date of creation	2013-03-22 13:03:42
Last modified on	2013-03-22 13:03:42
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	8
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 17B99