quadratic Lie algebra

A Lie algebraMathworldPlanetmath 𝔤 is said to be quadratic if 𝔤 as a representationPlanetmathPlanetmath (under the adjoint action) admits a non-degenerate, invariant scalar product () .

𝔤 being quadratic implies that the adjointPlanetmathPlanetmathPlanetmathPlanetmath and co-adjoint representations of 𝔤 are isomorphicPlanetmathPlanetmathPlanetmath.

Indeed, the non-degeneracy of () implies that the induced map ϕ:𝔤𝔤* given by ϕ(X)(Z)=(XZ) is an isomorphismMathworldPlanetmathPlanetmath of vector spacesMathworldPlanetmath. Invariance of the scalar productMathworldPlanetmath means that ([X,Y]Z)=-(Y[X,Z])=(Y[Z,X]). This implies that ϕ is a map of representations:

Title quadratic Lie algebra
Canonical name QuadraticLieAlgebra
Date of creation 2013-03-22 15:30:44
Last modified on 2013-03-22 15:30:44
Owner benjaminfjones (879)
Last modified by benjaminfjones (879)
Numerical id 6
Author benjaminfjones (879)
Entry type Definition
Classification msc 17B10
Classification msc 17B01
Related topic quadraticAlgebra
Defines quadratic Lie algebra