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.


  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


    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