index of a Lie algebra

Let 𝔮 be a Lie algebraMathworldPlanetmath over 𝕂 and 𝔮* its vector space dual. For ξ𝔮* let 𝔮ξ denote the stabilizerMathworldPlanetmath of ξ with respect to the co-adjoint representationPlanetmathPlanetmath.

The index of 𝔮 is defined to be



  1. 1.

    If 𝔮 is reductive then ind𝔮=rank𝔮. Indeed, 𝔮 and 𝔮* are isomorphicPlanetmathPlanetmathPlanetmath as representations for 𝔮 and so the index is the minimalPlanetmathPlanetmath dimension among stabilizers of elements in 𝔮. In particular the minimum is realized in the stabilizer of any regularPlanetmathPlanetmathPlanetmath element of 𝔮. These elemtents have stabilizer dimension equal to the rank of 𝔮.

  2. 2.

    If ind𝔮=0 then 𝔮 is called a Frobenius Lie algebra. This is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to condition that the Kirillov form Kξ:𝔮×𝔮𝕂 given by (X,Y)ξ([X,Y]) is non-singular for some ξ𝔮*. Another equivalent condition when 𝔮 is the Lie algebra of an algebraic group Q is that 𝔮 is Frobenius if and only if Q has an open orbit on 𝔮*.

Title index of a Lie algebra
Canonical name IndexOfALieAlgebra
Date of creation 2013-03-22 15:30:47
Last modified on 2013-03-22 15:30:47
Owner benjaminfjones (879)
Last modified by benjaminfjones (879)
Numerical id 6
Author benjaminfjones (879)
Entry type Definition
Classification msc 17B05
Defines index of a Lie algebra
Defines Frobenius Lie algebra
Defines Kirillov form