Regular element of a Lie algebra

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

Basic case

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

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

For a connected compact Lie group G {\displaystyle G} {\displaystyle G}, the regular elements form an open dense subset, made up of G {\displaystyle G} {\displaystyle G}-conjugacy classes of the elements in a maximal torus T {\displaystyle T} {\displaystyle T} which are regular in G {\displaystyle G} {\displaystyle G}. The regular elements of T {\displaystyle T} {\displaystyle T} are themselves explicitly given as the complement of a set in T {\displaystyle T} {\displaystyle T}, a set of codimension-one subtori corresponding to the root system of G {\displaystyle G} {\displaystyle G}. Similarly, in the Lie algebra g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} of G {\displaystyle G} {\displaystyle G}, the regular elements form an open dense subset which can be described explicitly as adjoint G {\displaystyle G} {\displaystyle G}-orbits of regular elements of the Lie algebra of T {\displaystyle T} {\displaystyle T}, the elements outside the hyperplanes corresponding to the root system.[1]

Definition

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

p x ( t ) = det ( t − ad ⁡ ( x ) ) = ∑ i = 0 dim ⁡ g a i ( x ) t i {\displaystyle p_{x}(t)=\det(t-\operatorname {ad} (x))=\sum _{i=0}^{\dim {\mathfrak {g}}}a_{i}(x)t^{i}} {\displaystyle p_{x}(t)=\det(t-\operatorname {ad} (x))=\sum _{i=0}^{\dim {\mathfrak {g}}}a_{i}(x)t^{i}}

be the characteristic polynomial of the adjoint endomorphism ad ⁡ ( x ) : y ↦ [ x , y ] {\displaystyle \operatorname {ad} (x):y\mapsto [x,y]} {\displaystyle \operatorname {ad} (x):y\mapsto [x,y]} of g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}}. Then, by definition, the rank of g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} is the least integer r {\displaystyle r} {\displaystyle r} such that a r ( x ) ≠ 0 {\displaystyle a_{r}(x)\neq 0} {\displaystyle a_{r}(x)\neq 0} for some x ∈ g {\displaystyle x\in {\mathfrak {g}}} {\displaystyle x\in {\mathfrak {g}}} and is denoted by rk ⁡ ( g ) {\displaystyle \operatorname {rk} ({\mathfrak {g}})} {\displaystyle \operatorname {rk} ({\mathfrak {g}})}.[3] For example, since a dim ⁡ g ( x ) = 1 {\displaystyle a_{\dim {\mathfrak {g}}}(x)=1} {\displaystyle a_{\dim {\mathfrak {g}}}(x)=1} for every x, g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} is nilpotent (i.e., each ad ⁡ ( x ) {\displaystyle \operatorname {ad} (x)} {\displaystyle \operatorname {ad} (x)} is nilpotent by Engel's theorem) if and only if rk ⁡ ( g ) = dim ⁡ g {\displaystyle \operatorname {rk} ({\mathfrak {g}})=\dim {\mathfrak {g}}} {\displaystyle \operatorname {rk} ({\mathfrak {g}})=\dim {\mathfrak {g}}}. Note that over an algebraically closed field of characteristic zero and h {\displaystyle {\mathfrak {h}}} {\displaystyle {\mathfrak {h}}} is a Cartan subalgebra of g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} then rk ⁡ ( g ) = dim ⁡ h {\displaystyle \operatorname {rk} ({\mathfrak {g}})=\dim {\mathfrak {h}}} {\displaystyle \operatorname {rk} ({\mathfrak {g}})=\dim {\mathfrak {h}}}; see the section below.

Let g reg = { x ∈ g | a rk ⁡ ( g ) ( x ) ≠ 0 } {\displaystyle {\mathfrak {g}}_{\text{reg}}=\{x\in {\mathfrak {g}}|a_{\operatorname {rk} ({\mathfrak {g}})}(x)\neq 0\}} {\displaystyle {\mathfrak {g}}_{\text{reg}}=\{x\in {\mathfrak {g}}|a_{\operatorname {rk} ({\mathfrak {g}})}(x)\neq 0\}}. By definition, a regular element of g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} is an element of the set g reg {\displaystyle {\mathfrak {g}}_{\text{reg}}} {\displaystyle {\mathfrak {g}}_{\text{reg}}}.[3] Since a rk ⁡ ( g ) {\displaystyle a_{\operatorname {rk} ({\mathfrak {g}})}} {\displaystyle a_{\operatorname {rk} ({\mathfrak {g}})}} is a polynomial function on g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}}, with respect to the Zariski topology, the set g reg {\displaystyle {\mathfrak {g}}_{\text{reg}}} {\displaystyle {\mathfrak {g}}_{\text{reg}}} is an open subset of g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}}.

Over C {\displaystyle \mathbb {C} } {\displaystyle \mathbb {C} }, g reg {\displaystyle {\mathfrak {g}}_{\text{reg}}} {\displaystyle {\mathfrak {g}}_{\text{reg}}} is a connected set (with respect to the usual topology),[4] but over R {\displaystyle \mathbb {R} } {\displaystyle \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 ∈ g {\displaystyle x\in {\mathfrak {g}}} {\displaystyle x\in {\mathfrak {g}}}, let

g 0 ( x ) = ∑ n ≥ 0 ker ⁡ ( ad ⁡ ( x ) n : g → g ) {\displaystyle {\mathfrak {g}}^{0}(x)=\sum _{n\geq 0}\ker(\operatorname {ad} (x)^{n}:{\mathfrak {g}}\to {\mathfrak {g}})} {\displaystyle {\mathfrak {g}}^{0}(x)=\sum _{n\geq 0}\ker(\operatorname {ad} (x)^{n}:{\mathfrak {g}}\to {\mathfrak {g}})}

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

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

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

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

For a Cartan subalgebra h {\displaystyle {\mathfrak {h}}} {\displaystyle {\mathfrak {h}}} of a complex semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} {\displaystyle {\mathfrak {g}}} with the root system Φ {\displaystyle \Phi } {\displaystyle \Phi }, an element of h {\displaystyle {\mathfrak {h}}} {\displaystyle {\mathfrak {h}}} is regular if and only if it is not in the union of hyperplanes ⋃ α ∈ Φ { h ∈ h ∣ α ( h ) = 0 } {\textstyle \bigcup _{\alpha \in \Phi }\{h\in {\mathfrak {h}}\mid \alpha (h)=0\}} {\textstyle \bigcup _{\alpha \in \Phi }\{h\in {\mathfrak {h}}\mid \alpha (h)=0\}}.[10] This is because: for r = dim ⁡ h {\displaystyle r=\dim {\mathfrak {h}}} {\displaystyle r=\dim {\mathfrak {h}}},

  • For each h ∈ h {\displaystyle h\in {\mathfrak {h}}} {\displaystyle h\in {\mathfrak {h}}}, the characteristic polynomial of ad ⁡ ( h ) {\displaystyle \operatorname {ad} (h)} {\displaystyle \operatorname {ad} (h)} is t r ( t dim ⁡ g − r − ∑ α ∈ Φ α ( h ) t dim ⁡ g − r − 1 + ⋯ ± ∏ α ∈ Φ α ( h ) ) {\textstyle 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)} {\textstyle 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)}.

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

Notes

  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.
  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.
  9. Bourbaki 1981, Ch. VII, § 3.3. Theorem 2.
  10. Procesi 2007, Ch. 10, § 3.2.

References