regular element of a Lie algebra

An element $X\in\mathfrak{g}$ of a Lie algebra is called regular if the dimension of its centralizer $\zeta_{\mathfrak{g}}(X)=\{Y\in\mathfrak{g}\mid[X,Y]=0\}$ is minimal among all centralizers of elements in $\mathfrak{g}$ .

Regular elements clearly exist and moreover they are Zariski dense in $\mathfrak{g}$ . The function $X\mapsto\dim\zeta_{\mathfrak{g}}(X)$ is an upper semi-continuous function $\mathfrak{g}\to\mathbb{Z}_{\geq 0}$ . Indeed, it is a constant minus $\operatorname{rank}(ad_{X})$ and $X\mapsto\operatorname{rank}(ad_{X})$ 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 $\mathfrak{g}$ is reductive then the minimal centralizer dimension is equal to the rank of $\mathfrak{g}$ .

More generally if $V$ is a representation for a Lie algebra $\mathfrak{g}$ , 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 $\mathfrak{sl}_{n}\mathbb{C}$ a diagonal matrix $X=\operatorname{diag}(s_{1},\ldots,s_{n})$ is regular iff $(s_{i}-s_{j})\neq 0$ for all pairs $1\leq i<j\leq n$ . Any conjugate of such a matrix is also obviously regular.
2.

In $\mathfrak{sl}_{n}\mathbb{C}$ the nilpotent matrix

$\left(\begin{array}[]{ccccc}0&1&\cdots&&0\\ 0&0&1&&\\ \vdots&&\ddots&\ddots&1\\ 0&&\cdots&&0\end{array}\right)$

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	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