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 formula_1 matrices over an algebraically closed field (such as the complex numbers), an element formula_2 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 formula_3 evaluated at the matrix formula_2, and therefore the centralizer has dimension formula_3 (but it is not necessarily an algebraic torus).
If the matrix formula_2 is diagonalisable, then it is regular if and only if there are formula_3 different eigenvalues. To see this, notice that formula_2 will commute with any matrix formula_9 that stabilises each of its eigenspaces. If there are formula_3 different eigenvalues, then this happens only if formula_9 is diagonalisable on the same basis as formula_2; in fact formula_9 is a linear combination of the first formula_3 powers of formula_2, and the centralizer is an algebraic torus of complex dimension formula_3 (real dimension formula_17); since this is the smallest possible dimension of a centralizer, the matrix formula_2 is regular. However if there are equal eigenvalues, then the centralizer is the product of the general linear groups of the eigenspaces of formula_2, and has strictly larger dimension, so that formula_2 is not regular. 
For a connected compact Lie group formula_21, the regular elements form an open dense subset, made up of formula_21-conjugacy classes of the elements in a maximal torus formula_23 which are regular in formula_21. The regular elements of formula_23 are themselves explicitly given as the complement of a set in formula_23, a set of codimension-one subtori corresponding to the root system of formula_21. Similarly, in the Lie algebra formula_28 of formula_21, the regular elements form an open dense subset which can be described explicitly as adjoint formula_21-orbits of regular elements of the Lie algebra of formula_23, the elements outside the hyperplanes corresponding to the root system.
Definition.
Let formula_28 be a finite-dimensional Lie algebra over an infinite field. For each formula_33, let
be the characteristic polynomial of the adjoint endomorphism formula_35 of formula_36. Then, by definition, the rank of formula_28 is the least integer formula_38 such that formula_39 for some formula_40 and is denoted by formula_41. For example, since formula_42 for every "x", formula_36 is nilpotent (i.e., each formula_44 is nilpotent by Engel's theorem) if and only if formula_45.
Let formula_46. By definition, a regular element of formula_28 is an element of the set formula_48. Since formula_49 is a polynomial function on formula_28, with respect to the Zariski topology, the set formula_48 is an open subset of formula_28.
Over formula_53, formula_48 is a connected set (with respect to the usual topology), but over formula_55, 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 formula_33, let
be the generalized eigenspace of formula_44 for eigenvalue zero. It is a subalgebra of formula_36. Note that formula_60 is the same as the (algebraic) multiplicity of zero as an eigenvalue of formula_44; i.e., the least integer "m" such that formula_62 in the notation in . Thus, formula_63 and the equality holds if and only if formula_64 is a regular element.
The statement is then that if formula_64 is a regular element, then formula_66 is a Cartan subalgebra. Thus, formula_67 is the dimension of at least some Cartan subalgebra; in fact, formula_67 is the minimum dimension of a Cartan subalgebra. More strongly, over a field of characteristic zero (e.g., formula_55 or formula_53),
A regular element in a Cartan subalgebra of a complex semisimple Lie algebra.
For a Cartan subalgebra formula_77 of a complex semisimple Lie algebra formula_36 with the root system formula_79, an element of formula_77 is regular if and only if it is not in the union of hyperplanes formula_81. This is because: for formula_82,
This characterization is sometimes taken as the definition of a regular element (especially when only regular elements in Cartan subalgebras are of interest).