Regular element of a Lie algebra
Encyclopedia
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 the complex general linear group of nxn matrices, a diagonal matrix M will commute with its powers, and so with convergent series in those powers. If the diagonal entries are distinct and "sufficiently general", that is all, and such elements are regular. The centralizer is the algebraic torus
generated by M, of dimension 2n as a real manifold.
For a connected compact Lie group G, and its Lie algebra g, the regular elements can also be described explicitly. In g they form an open and dense subset. In G, the regular elements form an open dense subset also; and if T is a maximal torus
of G, the elements t of T that are regular in G determine the regular elements of G, which make up the union of the conjugacy class
es in G of regular elements in T. The regular elements t are themselves explicitly given as the complement of a set in T, determined by the adjoint action of G, and making up a union of subtori.
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
or Lie group
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
is an element whose centralizer has dimension as small as possible. For example, in the complex general linear group of nxn matrices, a diagonal matrix M will commute with its powers, and so with convergent series in those powers. If the diagonal entries are distinct and "sufficiently general", that is all, and such elements are regular. The centralizer is the algebraic torus
Algebraic torus
In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory...
generated by M, of dimension 2n as a real manifold.
For a connected compact Lie group G, and its Lie algebra g, the regular elements can also be described explicitly. In g they form an open and dense subset. In G, the regular elements form an open dense subset also; and if T is a maximal torus
Maximal torus
In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups.A torus in a Lie group G is a compact, connected, abelian Lie subgroup of G . A maximal torus is one which is maximal among such subgroups...
of G, the elements t of T that are regular in G determine the regular elements of G, which make up the union of the conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
es in G of regular elements in T. The regular elements t are themselves explicitly given as the complement of a set in T, determined by the adjoint action of G, and making up a union of subtori.