regular element of a Lie algebra
An element $X\in \U0001d524$ of a Lie algebra^{} is called regular^{} if the dimension^{} of its centralizer^{} ${\zeta}_{\U0001d524}(X)=\{Y\in \U0001d524\mid [X,Y]=0\}$ is minimal among all centralizers of elements in $\U0001d524$.
Regular elements clearly exist and moreover they are Zariski dense in $\U0001d524$. The function $X\mapsto dim{\zeta}_{\U0001d524}(X)$ is an upper semicontinuous function $\U0001d524\to {\mathbb{Z}}_{\ge 0}$. Indeed, it is a constant minus $\mathrm{rank}(a{d}_{X})$ and $X\mapsto \mathrm{rank}(a{d}_{X})$ is lower semicontinuous. Thus the set of elements whose centralizer dimension is (greater than or) equal to that of any given regular element is Zariski open and nonempty.
If $\U0001d524$ is reductive then the minimal centralizer dimension is equal to the rank of $\U0001d524$.
More generally if $V$ is a representation for a Lie algebra $\U0001d524$, an element $v\in V$ is called regular if the dimension of its stabilizer^{} is minimal among all stabilizers of elements in $V$.
Examples

1.
In $\U0001d530{\U0001d529}_{n}\u2102$ a diagonal matrix^{} $X=\mathrm{diag}({s}_{1},\mathrm{\dots},{s}_{n})$ is regular iff $({s}_{i}{s}_{j})\ne 0$ for all pairs $$. Any conjugate^{} of such a matrix is also obviously regular.

2.
In $\U0001d530{\U0001d529}_{n}\u2102$ the nilpotent matrix^{}
$$\left(\begin{array}{ccccc}\hfill 0\hfill & \hfill 1\hfill & \hfill \mathrm{\cdots}\hfill & & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & & \\ \hfill \mathrm{\vdots}\hfill & & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill 1\hfill \\ \hfill 0\hfill & & \hfill \mathrm{\cdots}\hfill & & \hfill 0\hfill \end{array}\right)$$ is regular. Moreover, it’s adjoint^{} orbit contains the set of all regular nilpotent elements^{}. 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  20130322 15:30:53 
Last modified on  20130322 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 