regular element of a Lie algebra


An element X∈𝔤 of a Lie algebraMathworldPlanetmath is called regularPlanetmathPlanetmathPlanetmath if the dimensionPlanetmathPlanetmathPlanetmath of its centralizerMathworldPlanetmathPlanetmath ζ𝔤⁢(X)={Y∈𝔤∣[X,Y]=0} is minimal among all centralizers of elements in 𝔤.

Regular elements clearly exist and moreover they are Zariski dense in 𝔤. The function X↦dim⁡ζ𝔤⁢(X) is an upper semi-continuous function 𝔤→ℤ≥0. Indeed, it is a constant minus rank⁡(a⁢dX) and X↦rank⁡(a⁢dX) is lower semi-continuous. Thus the set of elements whose centralizer dimension is (greater than or) equal to that of any given regular element is Zariski open and non-empty.

If 𝔤 is reductive then the minimal centralizer dimension is equal to the rank of 𝔤.

More generally if V is a representation for a Lie algebra 𝔤, an element v∈V is called regular if the dimension of its stabilizerMathworldPlanetmath is minimal among all stabilizers of elements in V.

Examples

  1. 1.

    In 𝔰⁢𝔩n⁢ℂ a diagonal matrixMathworldPlanetmath X=diag⁡(s1,…,sn) is regular iff (si-sj)≠0 for all pairs 1≤i<j≤n. Any conjugatePlanetmathPlanetmath of such a matrix is also obviously regular.

  2. 2.

    In 𝔰⁢𝔩n⁢ℂ the nilpotent matrixMathworldPlanetmath

    (01⋯0001⋮⋱⋱10⋯0)

    is regular. Moreover, it’s adjointPlanetmathPlanetmathPlanetmath orbit contains the set of all regular nilpotent elementsMathworldPlanetmath. The centralizer of this matrix is the full subalgebra of trace zero, diagonal matricies.

Title regular element of a Lie algebra
Canonical name RegularElementOfALieAlgebra
Date of creation 2013-03-22 15:30:53
Last modified on 2013-03-22 15:30:53
Owner benjaminfjones (879)
Last modified by benjaminfjones (879)
Numerical id 6
Author benjaminfjones (879)
Entry type Definition
Classification msc 17B05
Defines regular element