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 formula_1 is regular if its centralizer in formula_2 has dimension equal to the rank of formula_2, which in turn equals the dimension of some Cartan subalgebra formula_4 (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 formula_5  a Lie group is regular if its centralizer has dimension equal to the rank of formula_6. Basic case. In the specific case of formula_7, the Lie algebra of formula_8 matrices over an algebraically closed field formula_9(such as the complex numbers), a regular element formula_10 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 formula_11 evaluated at the matrix formula_10, and therefore the centralizer has dimension formula_11 (which equals  the rank of formula_14, but is not necessarily an algebraic torus). If the matrix formula_10 is diagonalisable, then it is regular if and only if there are formula_11 different eigenvalues. To see this, notice that formula_10 will commute with any matrix formula_18 that stabilises each of its eigenspaces. If there are formula_11 different eigenvalues, then this happens only if formula_18 is diagonalisable on the same basis as formula_10; in fact formula_18 is a linear combination of the first formula_11 powers of formula_10, and the centralizer is an algebraic torus of complex dimension formula_11 (real dimension formula_26); since this is the  smallest possible dimension of a centralizer, the matrix formula_10 is regular. However if there are equal eigenvalues, then the centralizer is the product of the general linear groups of the eigenspaces of formula_10, and has strictly larger dimension, so that formula_10 is not regular. For a connected compact Lie group formula_30, the regular elements form an open dense subset, made up of formula_30-conjugacy classes of the elements in a maximal torus formula_32 which are regular in formula_30. The regular elements of formula_32 are themselves explicitly given as the complement of a set in formula_32, a set of codimension-one  subtori corresponding to the root system of formula_30. Similarly, in the Lie algebra formula_2 of formula_30, the regular elements form an open dense subset which can be described explicitly as adjoint formula_30-orbits of regular elements of the Lie algebra of formula_32, the elements outside the hyperplanes corresponding to the root system. Definition. Let formula_2 be a finite-dimensional Lie algebra over an infinite field. For each formula_42, let be the characteristic polynomial of the adjoint endomorphism formula_44 of formula_45. Then, by definition, the rank of formula_2 is the least integer formula_47 such that formula_48 for some formula_49 and is  denoted by formula_50. For example, since formula_51 for every "x", formula_45 is nilpotent (i.e., each formula_53 is nilpotent by Engel's theorem) if and only if formula_54. Let formula_55. By definition, a regular element of formula_2 is an element of the set formula_57. Since formula_58 is a polynomial function on formula_2, with respect to the Zariski topology, the set formula_57 is an open subset of formula_2. Over formula_62, formula_57 is a connected set (with respect to the usual topology), but over formula_64, 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_42, let be the generalized eigenspace of formula_53 for eigenvalue zero. It is a subalgebra of formula_45. Note that formula_69 is the same as the (algebraic) multiplicity of zero as an eigenvalue of formula_53; i.e., the least integer "m" such that formula_71 in the notation in . Thus, formula_72 and the equality holds if and only if formula_73 is a regular element. The statement is then  that if formula_73 is a regular element, then formula_75 is a Cartan subalgebra. Thus, formula_76 is the dimension of at least some Cartan subalgebra; in fact, formula_76 is the minimum dimension of a Cartan subalgebra. More strongly, over a field of characteristic zero (e.g., formula_64 or formula_62), A regular element in a Cartan subalgebra of a complex semisimple Lie algebra. For a Cartan subalgebra formula_86 of a complex semisimple Lie algebra formula_45 with the root system formula_88, an element of formula_86 is regular if and only if it is not in the union of hyperplanes formula_90. This is because: for  formula_91, This characterization is sometimes taken as the definition of a regular element (especially when only regular elements in Cartan subalgebras are of interest).