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.

Basic case

In the specific case of $n\times n$ matrices over an algebraically closed field (such as the complex numbers), an element $M$ is regular if and only if its Jordan normal form contains a single Jordan block for each eigenvalue. In that case, the centralizer is the set of polynomials of degree less than $n$ evaluated at the matrix $M$ , and therefore the centralizer has dimension $n$ (but it 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.^[1]

Definition

Let ${\mathfrak {g}}$ be a finite-dimensional Lie algebra over an infinite field.^[2] For each $x\in {\mathfrak {g}}$ , let

p_{x}(t)=\operatorname {det} (t-\operatorname {ad} (x))=\sum _{i=0}^{\dim {\mathfrak {g}}}a_{i}(x)t^{i}

be the characteristic polynomial of the adjoint endomorphism $\operatorname {ad} (x):y\mapsto [x,y]$ of ${\mathfrak {g}}$ . Then, by definition, the rank of ${\mathfrak {g}}$ is the least integer $r$ such that $a_{r}(x)\neq 0$ for some $x\in {\mathfrak {g}}$ and is denoted by $\operatorname {rk} ({\mathfrak {g}})$ .^[3] For example, since $a_{\dim {\mathfrak {g}}}(x)=1$ for every x, ${\mathfrak {g}}$ is nilpotent (i.e., each $\operatorname {ad} (x)$ is nilpotent by Engel's theorem) if and only if $\operatorname {rk} ({\mathfrak {g}})=\dim {\mathfrak {g}}$ .

Let ${\mathfrak {g}}_{\text{reg}}=\{x\in {\mathfrak {g}}|a_{\operatorname {rk} ({\mathfrak {g}})}(x)\neq 0\}$ . By definition, a regular element of ${\mathfrak {g}}$ is an element of the set ${\mathfrak {g}}_{\text{reg}}$ .^[3] Since $a_{\operatorname {rk} ({\mathfrak {g}})}$ 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),^[4] but over $\mathbb {R}$ , it is only a finite union of connected open sets.^[5]

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)=\cup _{n\geq 0}\operatorname {ker} (\operatorname {ad} (x)^{n}:{\mathfrak {g}}\to {\mathfrak {g}})

be the generalized eigenspace of $\operatorname {ad} (x)$ for eigenvalue zero. It is a subalgebra of ${\mathfrak {g}}$ .^[6] Note that $\dim {\mathfrak {g}}^{0}(x)$ is the same as the (algebraic) multiplicity^[7] of zero as an eigenvalue of $\operatorname {ad} (x)$ ; i.e., the least integer m such that $a_{m}(x)\neq 0$ in the notation in #Definition. Thus, $\operatorname {rk} ({\mathfrak {g}})\leq \dim {\mathfrak {g}}^{0}(x)$ and the equality holds if and only if $x$ is a regular element.^[3]

The statement is then that if $x$ is a regular element, then ${\mathfrak {g}}^{0}(x)$ is a Cartan subalgebra.^[8] Thus, $\operatorname {rk} ({\mathfrak {g}})$ is the dimension of at least some Cartan subalgebra; in fact, $\operatorname {rk} ({\mathfrak {g}})$ is the minimum dimension of a Cartan subalgebra. More strongly, over a field of characteristic zero (e.g., $\mathbb {R}$ or $\mathbb {C}$ ),^[9]

every Cartan subalgebra of ${\mathfrak {g}}$ has the same dimension; thus, $\operatorname {rk} ({\mathfrak {g}})$ 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 $\Phi$ , an element of ${\mathfrak {h}}$ is regular if and only if it is not in the union of hyperplanes $\bigcup _{\alpha \in \Phi }\{h\in {\mathfrak {h}}|\alpha (h)=0\}$ .^[10] This is because: for $r=\dim {\mathfrak {h}}$ ,

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

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

Notes

^ Sepanski, Mark R. (2006). Compact Lie Groups. Springer. p. 156. ISBN 978-0-387-30263-8.
^ Editorial note: the definition of a regular element over a finite field is unclear.
^ ^a ^b ^c Bourbaki 1981, Ch. VII, § 2.2. Definition 2.
^ Serre 2001, Ch. III, § 1. Proposition 1.
^ Serre 2001, Ch. III, § 6.
^ This is a consequence of the binomial-ish formula for ad.
^ Recall that the geometric multiplicity of an eigenvalue of an endomorphism is the dimension of the eigenspace while the algebraic multiplicity of it is the dimension of the generalized eigenspace.
^ Bourbaki 1981, Ch. VII, § 2.3. Theorem 1.
^ Bourbaki 1981, Ch. VII, § 3.3. Theorem 2.
^ Procesi 2001, Ch. 10, § 3.2.

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, vol. 129, Berlin, New York: Springer-Verlag, ISBN 978-0-387-97495-8, MR 1153249
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

[1] Sepanski, Mark R. (2006). Compact Lie Groups. Springer. p. 156. ISBN 978-0-387-30263-8.

[2] Editorial note: the definition of a regular element over a finite field is unclear.

[Def-3] Bourbaki 1981, Ch. VII, § 2.2. Definition 2.

[4] Serre 2001, Ch. III, § 1. Proposition 1.

[5] Serre 2001, Ch. III, § 6.

[6] This is a consequence of the binomial-ish formula for ad.

[7] Recall that the geometric multiplicity of an eigenvalue of an endomorphism is the dimension of the eigenspace while the algebraic multiplicity of it is the dimension of the generalized eigenspace.

[8] Bourbaki 1981, Ch. VII, § 2.3. Theorem 1.

[application-9] Bourbaki 1981, Ch. VII, § 3.3. Theorem 2.

[10] Procesi 2001, Ch. 10, § 3.2.

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