JacobiIdentityInterpretations 2002-09-20 11:52:22 2007-04-01 00:26:48 Definition Lie algebra \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \newcommand{\ad}{\operatorname{ad}} \newcommand{\End}{\operatorname{End}} 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 \footnote{Here, ``$\mathfrak{gl}(\mathfrak{g})$'' means the space o endomorphisms of $\mathfrak{g}$, viewed as a vector space, with Lie bracket on $\mathfrak{gl}(\mathfrak{g})$being commutator.} $\ad:\mathfrak{g} \rightarrow \mathfrak{gl}(\mathfrak{g})$ really is a representation. Yet another way to formulate the identity is \[ \ad(x)[y,z]=[\ad(x)y,z]+[y,\ad(x)z], \] i.e., $\ad(x)$ is a derivation on $\mathfrak{g}$ for all $x \in \mathfrak{g}$.