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...

(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...

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...

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

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...

(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...

defined as the unit group of the matrix ring M(n, R). The general linear group GL(n, R) over the field of real numbers is a real Lie group of dimension n2...

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

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;...

The Lorentz group is a Lie group of symmetries of the spacetime of special relativity. This group can be realized as a collection of matrices, linear...

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

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

this group. The unitary group U(n) is a real Lie group of dimension n2. The Lie algebra of U(n) consists of n × n skew-Hermitian matrices, with the Lie bracket...

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

symplectic group over the field of complex numbers is a non-compact, simply connected, simple Lie group. Sp(n, C) is the complexification of the real group Sp(2n...

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

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...

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...

non-abelian Lie group that is of importance as a model in our understanding of the most basic fundamentals of physics. The Poincaré group consists of...

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...

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

and orientation. When F is R or C, SL(n, F) is a Lie subgroup of GL(n, F) of dimension n2 − 1. The Lie algebra s l ( n , F ) {\displaystyle {\mathfrak...

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...

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...

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...

particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system...

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...

this passage from real to complexified Lie algebra is harmless. The reason for passing to the complexification is that it allows us to construct a nice...