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
, which in turn equals the dimension of some Cartan subalgebra
(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).
a Lie group is regular if its centralizer has dimension equal to the rank of
matrices over an algebraically closed field
(such as the complex numbers), a regular element
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
, and the centralizer is an algebraic torus of complex dimension
); since this is the smallest possible dimension of a centralizer, the matrix
However if there are equal eigenvalues, then the centralizer is the product of the general linear groups of the eigenspaces of
For a connected compact Lie group
, the regular elements form an open dense subset, made up of
-conjugacy classes of the elements in a maximal torus
, a set of codimension-one subtori corresponding to the root system of
, the regular elements form an open dense subset which can be described explicitly as adjoint
-orbits of regular elements of the Lie algebra of
, the elements outside the hyperplanes corresponding to the root system.
be a finite-dimensional Lie algebra over an infinite field.
, let be the characteristic polynomial of the adjoint endomorphism
, with respect to the Zariski topology, the set
is a connected set (with respect to the usual topology),[4] but over
, it is only a finite union of connected open sets.
[5] 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.
is the dimension of at least some Cartan subalgebra; in fact,
is the minimum dimension of a Cartan subalgebra.
of a complex semisimple Lie algebra
, This characterization is sometimes taken as the definition of a regular element (especially when only regular elements in Cartan subalgebras are of interest).