Regular element of a Lie algebra

In mathematics, a regular element of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie algebra, an element $X\in \mathfrak{𝔤}$ is regular if its centralizer in $\mathfrak{𝔤}$ has dimension equal to the rank of $\mathfrak{𝔤}$, which in turn equals the dimension of some Cartan subalgebra $\mathfrak{𝔥}$ (note that in earlier papers, an element of a complex semisimple Lie algebra was termed regular if it is semisimple and the kernel of its adjoint representation is a Cartan subalgebra). An element $g\in G$ a Lie group is regular if its centralizer has dimension equal to the rank of $G$.

Basic case

In the specific case of ${\mathfrak{𝔤}\mathfrak{𝔩}}_n(\mathbb{𝕜})$, the Lie algebra of $n\times n$ matrices over an algebraically closed field $\mathbb{𝕜}$(such as the complex numbers), a regular element $M$ is an element whose Jordan normal form contains a single Jordan block for each eigenvalue (in other words, the geometric multiplicity of each eigenvalue is 1). The centralizer of a regular element is the set of polynomials of degree less than $n$ evaluated at the matrix $M$, and therefore the centralizer has dimension $n$ (which equals the rank of ${\mathfrak{𝔤}\mathfrak{𝔩}}_n$, but is not necessarily an algebraic torus).

If the matrix $M$ is diagonalisable, then it is regular if and only if there are $n$ different eigenvalues. To see this, notice that $M$ will commute with any matrix $P$ that stabilises each of its eigenspaces. If there are $n$ different eigenvalues, then this happens only if $P$ is diagonalisable on the same basis as $M$; in fact $P$ is a linear combination of the first $n$ powers of $M$, and the centralizer is an algebraic torus of complex dimension $n$ (real dimension $2n$); since this is the smallest possible dimension of a centralizer, the matrix $M$ is regular. However if there are equal eigenvalues, then the centralizer is the product of the general linear groups of the eigenspaces of $M$, and has strictly larger dimension, so that $M$ is not regular.

For a connected compact Lie group $G$, the regular elements form an open dense subset, made up of $G$-conjugacy classes of the elements in a maximal torus $T$ which are regular in $G$. The regular elements of $T$ are themselves explicitly given as the complement of a set in $T$, a set of codimension-one subtori corresponding to the root system of $G$. Similarly, in the Lie algebra $\mathfrak{𝔤}$ of $G$, the regular elements form an open dense subset which can be described explicitly as adjoint $G$-orbits of regular elements of the Lie algebra of $T$, the elements outside the hyperplanes corresponding to the root system.

Definition

Let $\mathfrak{𝔤}$ be a finite-dimensional Lie algebra over an infinite field. For each $x\in \mathfrak{𝔤}$, let

$p_x(t)=\det (t-\mathrm{ad}(x))=\sum _{i=0}^{\dim \mathfrak{𝔤}}{a}_i(x)t^i$

be the characteristic polynomial of the adjoint endomorphism $\mathrm{ad}(x):y\mapsto [x,y]$ of $\mathfrak{𝔤}$. Then, by definition, the rank of $\mathfrak{𝔤}$ is the least integer $r$ such that $a_r(x)\ne 0$ for some $x\in \mathfrak{𝔤}$ and is denoted by $\mathrm{rk}(\mathfrak{𝔤})$. For example, since $a_{\dim \mathfrak{𝔤}}(x)=1$ for every x, $\mathfrak{𝔤}$ is nilpotent (i.e., each $\mathrm{ad}(x)$ is nilpotent by Engel's theorem) if and only if $\mathrm{rk}(\mathfrak{𝔤})=\dim \mathfrak{𝔤}$.

Let ${\mathfrak{𝔤}}_{\text{reg}}=\{x\in \mathfrak{𝔤}|a_{\mathrm{rk}(\mathfrak{𝔤})}(x)\ne 0\}$. By definition, a regular element of $\mathfrak{𝔤}$ is an element of the set ${\mathfrak{𝔤}}_{\text{reg}}$. Since $a_{\mathrm{rk}(\mathfrak{𝔤})}$ is a polynomial function on $\mathfrak{𝔤}$, with respect to the Zariski topology, the set ${\mathfrak{𝔤}}_{\text{reg}}$ is an open subset of $\mathfrak{𝔤}$.

Over $\mathbb{ℂ}$, ${\mathfrak{𝔤}}_{\text{reg}}$ is a connected set (with respect to the usual topology), but over $\mathbb{ℝ}$, it is only a finite union of connected open sets.

A Cartan subalgebra and a regular element

Over an infinite field, a regular element can be used to construct a Cartan subalgebra, a self-normalizing nilpotent subalgebra. Over a field of characteristic zero, this approach constructs all the Cartan subalgebras.

Given an element $x\in \mathfrak{𝔤}$, let

${\mathfrak{𝔤}}^0(x)=\bigcup _{n\ge 0}\ker (\mathrm{ad}(x{)}^n:\mathfrak{𝔤}\to \mathfrak{𝔤})$

be the generalized eigenspace of $\mathrm{ad}(x)$ for eigenvalue zero. It is a subalgebra of $\mathfrak{𝔤}$. Note that $\dim {\mathfrak{𝔤}}^0(x)$ is the same as the (algebraic) multiplicity of zero as an eigenvalue of $\mathrm{ad}(x)$; i.e., the least integer m such that $a_m(x)\ne 0$ in the notation in § Definition. Thus, $\mathrm{rk}(\mathfrak{𝔤})\le \dim {\mathfrak{𝔤}}^0(x)$ and the equality holds if and only if $x$ is a regular element.

The statement is then that if $x$ is a regular element, then ${\mathfrak{𝔤}}^0(x)$ is a Cartan subalgebra. Thus, $\mathrm{rk}(\mathfrak{𝔤})$ is the dimension of at least some Cartan subalgebra; in fact, $\mathrm{rk}(\mathfrak{𝔤})$ is the minimum dimension of a Cartan subalgebra. More strongly, over a field of characteristic zero (e.g., $\mathbb{ℝ}$ or $\mathbb{ℂ}$),

• every Cartan subalgebra of $\mathfrak{𝔤}$ has the same dimension; thus, $\mathrm{rk}(\mathfrak{𝔤})$ is the dimension of an arbitrary Cartan subalgebra,
• an element x of $\mathfrak{𝔤}$ is regular if and only if ${\mathfrak{𝔤}}^0(x)$ is a Cartan subalgebra, and
• every Cartan subalgebra is of the form ${\mathfrak{𝔤}}^0(x)$ for some regular element $x\in \mathfrak{𝔤}$.

A regular element in a Cartan subalgebra of a complex semisimple Lie algebra

For a Cartan subalgebra $\mathfrak{𝔥}$ of a complex semisimple Lie algebra $\mathfrak{𝔤}$ with the root system $\mathrm{\Phi }$, an element of $\mathfrak{𝔥}$ is regular if and only if it is not in the union of hyperplanes $\bigcup _{\alpha \in \mathrm{\Phi }}\{h\in \mathfrak{𝔥}\mid \alpha (h)=0\}$. This is because: for $r=\dim \mathfrak{𝔥}$,

• For each $h\in \mathfrak{𝔥}$, the characteristic polynomial of $\mathrm{ad}(h)$ is $t^r(t^{\dim \mathfrak{𝔤}-r}-\sum _{\alpha \in \mathrm{\Phi }}\alpha (h)t^{\dim \mathfrak{𝔤}-r-1}+\cdots \pm \prod _{\alpha \in \mathrm{\Phi }}\alpha (h))$.

This characterization is sometimes taken as the definition of a regular element (especially when only regular elements in Cartan subalgebras are of interest).

References

• Bourbaki, N. (1981), Groupes et Algèbres de Lie, Éléments de Mathématique, Hermann 
• Fulton, William; Harris, Joe (1991), Representation Theory, A First Course, Graduate Texts in Mathematics, 129, Berlin, New York: Springer-Verlag, ISBN 978-0-387-97495-8 
• Procesi, Claudio (2007), Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402 
• Serre, Jean-Pierre (2001), Complex Semisimple Lie Algebras, Springer, ISBN 3-5406-7827-1