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 [math]\displaystyle{ X \in \mathfrak{g} }[/math] is regular if its centralizer in [math]\displaystyle{ \mathfrak{g} }[/math] has dimension equal to the rank of [math]\displaystyle{ \mathfrak{g} }[/math], which in turn equals the dimension of some Cartan subalgebra [math]\displaystyle{ \mathfrak{h} }[/math] (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 [math]\displaystyle{ g \in G }[/math] a Lie group is regular if its centralizer has dimension equal to the rank of [math]\displaystyle{ G }[/math].

Basic case

In the specific case of [math]\displaystyle{ \mathfrak{gl}_n(\mathbb{k}) }[/math], the Lie algebra of [math]\displaystyle{ n \times n }[/math] matrices over an algebraically closed field [math]\displaystyle{ \mathbb{k} }[/math](such as the complex numbers), a regular element [math]\displaystyle{ M }[/math] 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 [math]\displaystyle{ n }[/math] evaluated at the matrix [math]\displaystyle{ M }[/math], and therefore the centralizer has dimension [math]\displaystyle{ n }[/math] (which equals the rank of [math]\displaystyle{ \mathfrak{gl}_n }[/math], but is not necessarily an algebraic torus).

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

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

Definition

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

[math]\displaystyle{ p_x(t) = \det(t - \operatorname{ad}(x)) = \sum_{i=0}^{\dim \mathfrak{g}} a_i(x) t^i }[/math]

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

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

Over [math]\displaystyle{ \mathbb{C} }[/math], [math]\displaystyle{ \mathfrak{g}_{\text{reg}} }[/math] is a connected set (with respect to the usual topology), but over [math]\displaystyle{ \mathbb{R} }[/math], 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 [math]\displaystyle{ x \in \mathfrak{g} }[/math], let

[math]\displaystyle{ \mathfrak{g}^0(x) = \bigcup_{n \ge 0} \ker(\operatorname{ad}(x)^n : \mathfrak{g} \to \mathfrak{g}) }[/math]

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

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

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

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

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

• For each [math]\displaystyle{ h \in \mathfrak{h} }[/math], the characteristic polynomial of [math]\displaystyle{ \operatorname{ad}(h) }[/math] is [math]\displaystyle{ t^r \left(t^{\dim \mathfrak g - r} - \sum_{\alpha \in \Phi} \alpha(h) t^{\dim \mathfrak g - r - 1} + \cdots \pm \prod_{\alpha \in \Phi} \alpha(h)\right) }[/math].

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