# 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 JacobiIdentityInterpretations 2013-03-22 13:03:42 2013-03-22 13:03:42 rspuzio (6075) rspuzio (6075) 8 rspuzio (6075) Definition msc 17B99