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

and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0: g ≃ g 0 ⊗ R C . {\displaystyle...

complexification is a simple complex Lie algebra, unless L is already the complexification of a Lie algebra, in which case the complexification of L is a product of two...

(c-2)} -eigenspace. The Lie algebra s l ( 2 , C ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {C} )} is isomorphic to the complexification of s o ( 3 ) {\displaystyle...

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

group Complexification (Lie group) Simple Lie group Compact Lie group, Compact real form Semisimple Lie algebra Root system Simply laced group ADE classification...

compact Lie group, then the complexification of the Lie algebra of K is semisimple. Conversely, every complex semisimple Lie algebra has a compact real...

exceptional Lie algebras. If g 0 {\displaystyle {\mathfrak {g}}_{0}} is a finite-dimensional real simple Lie algebra, its complexification is either (1)...

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...

between compact connected Lie groups and complex reductive groups, up to isomorphism. For a compact Lie group K with complexification G, the inclusion from...

Lie algebra of a Lie group (or complexification of such), since, via the Lie correspondence, a Lie algebra representation can be integrated to a Lie group...

is the complexification of the real group Sp(2n, R). Sp(2n, R) is a real, non-compact, connected, simple Lie group. It has a fundamental group isomorphic...

In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for...

table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group (dimension; connectedness;...

pass to the complexification of the Lie algebra before proceeding.) To see how this works, consider the case G = SL(n, R). We can take the group of diagonal...

(G)=\operatorname {Lie} (K)\otimes _{\mathbb {R} }\mathbb {C} } , and (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example...

solvable Lie algebras are analogs of solvable groups. Any nilpotent Lie algebra is a fortiori solvable but the converse is not true. The solvable Lie algebras...

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...

say, a connected real semisimple linear Lie group G, then it has two natural actions: the complexification g {\displaystyle {\mathfrak {g}}} and the...

tori. A compact Lie algebra can be seen as the smallest real form of a corresponding complex Lie algebra, namely the complexification. Formally, one may...

\operatorname {GL} (n,\mathbb {R} )} over the field of real numbers is a real Lie group of dimension n 2 {\displaystyle n^{2}} . To see this, note that the set...

nilpotent Lie algebras are analogs of nilpotent groups. The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras,...

is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for...

said to be a real form of g {\displaystyle {\mathfrak {g}}} if the complexification g 0 ⊗ R C {\displaystyle {\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb...

subgroup). Every compact connected Lie group has a complexification, which is a complex reductive algebraic group. In fact, this construction gives a...

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...

mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same...

transformations. Mathematically, the Lorentz group may be described as the indefinite orthogonal group O(1, 3), the matrix Lie group that preserves the quadratic form...

representations. The spin group is used in physics when describing the symmetries of (electrically neutral, uncharged) fermions. Its complexification, Spinc, is used...

analytic continuation. Aut EC is the complexification of the compact Lie group Aut E in GL(EC). This follows because the Lie algebras of Aut EC and Aut E consist...