In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G...

simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be...

In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses...

matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn,...

fundamental group is trivial (denoted by 0). UC: If G is not simply connected, gives the universal cover of G. Note that a "complex Lie group" is defined...

universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that...

linear group over the field of complex numbers, GL ⁡ ( n , C ) {\displaystyle \operatorname {GL} (n,\mathbb {C} )} , is a complex Lie group of complex dimension...

connected Lie groups, the Lie group-Lie algebra correspondence is one-to-one. In this article, a Lie group refers to a real Lie group. For the complex and p-adic...

the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-dimensional Lie algebra over the real or complex numbers...

for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan. Using the Lie correspondence...

Lie group is a torus; i.e., a Lie group isomorphic to ( S 1 ) h {\displaystyle (S^{1})^{h}} . A connected complex Lie group that is a compact group is...

In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate...

unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may...

mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points...

Local Lie group Formal group law Hilbert's fifth problem Hilbert-Smith conjecture Lie group decompositions Real form (Lie theory) Complex Lie group Complexification...

equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components...

a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of...

connected Lie groups is the same as the classification of complex semisimple Lie algebras. Indeed, if K is a simply connected compact Lie group, then the...

is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimension 248. The complex Lie group E8 of complex dimension 248...

Lie subgroup Complex Lie group Local Lie group Poisson–Lie group Real Lie groups Simple Lie group Solvable Lie algebra Special linear Lie algebra Special...

of Lie groups, the exponential map is a map from the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} to the group, which...

mathematics, complex group may refer to: An archaic name for the symplectic group Complex reflection group A complex algebraic group A complex Lie group This...

complex group of complex dimension 133. The complex adjoint Lie group E7 of complex dimension 133 can be considered as a simple real Lie group of real dimension...

Lie groups using Weyl's unitary trick: each semisimple real Lie group G has a complexification, which is a complex Lie group Gc, and this complex Lie...

spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification...

M} . Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded...

representation theory of some complex Lie group G is in a qualitative way controlled by that of some compact real Lie group K, and the latter representation...

{\displaystyle {\mathfrak {g}}} is the Lie algebra of a complex Lie group, then a Borel subalgebra is the Lie algebra of a Borel subgroup. Let g = g l...

group G over a field k and closed subgroup H.[clarification needed] If X is a complex smooth projective variety and if G is a reductive complex Lie group...

complex semisimple Lie groups. These spaces can be obtained by taking a projectivization of a minimal nilpotent orbit of the respective complex Lie group...