Jacobi identity interpretations

The Jacobi identityMathworldPlanetmathPlanetmath in a Lie algebra 𝔤 has various interpretationsMathworldPlanetmathPlanetmath that are more transparent, whence easier to remember, than the usual form


One is the fact that the adjoint representationMathworldPlanetmath 11Here, “𝔤𝔩(𝔤)” means the space o endomorphismsPlanetmathPlanetmathPlanetmath of 𝔤, viewed as a vector space, with Lie bracket on 𝔤𝔩(𝔤)being commutatorPlanetmathPlanetmath. ad:𝔤𝔤𝔩(𝔤) really is a representationPlanetmathPlanetmath. Yet another way to formulate the identityPlanetmathPlanetmathPlanetmathPlanetmath is


i.e., ad(x) is a derivationPlanetmathPlanetmath on 𝔤 for all x𝔤.

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