of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra...

the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's...

concept in metaphysics, the theory suggests that the physical world is not as real or true as Forms. According to this theory, Forms—conventionally capitalized...

Lie group decompositions Real form (Lie theory) Complex Lie group Complexification (Lie group) Simple Lie group Compact Lie group, Compact real form Semisimple...

in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential...

second fundamental form), differential topology (intersection forms of manifolds, especially four-manifolds), Lie theory (the Killing form), and statistics...

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

group, Cartan subgroup, group of adjoint type, parabolic induction Real form (Lie theory), Satake diagram Adelic algebraic group, Weil's conjecture on Tamagawa...

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...

compact Lie group; this definition includes tori. Intrinsically and algebraically, a compact Lie algebra is a real Lie algebra whose Killing form is negative...

In the mathematical field of Lie theory, a split Lie algebra is a pair ( g , h ) {\displaystyle ({\mathfrak {g}},{\mathfrak {h}})} where g {\displaystyle...

geometry, a Lie-algebra-valued form is a differential form with values in a Lie algebra. Such forms have important applications in the theory of connections...

mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices...

and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach...

see real form for the case of real semisimple Lie algebras, which were classified by Élie Cartan. Further, the representation theory of semisimple Lie algebras...

of classifying the real simple Lie algebras to that of finding all the real forms of each complex simple Lie algebra (i.e., real Lie algebras whose complexification...

associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements...

its Lie algebra; this correspondence is discussed in detail in subsequent sections. See representation of Lie algebras for the Lie algebra theory. In...

conjugation. As well as the complex Lie group of type E8, there are three real forms of the Lie algebra, three real forms of the group with trivial center...

In mathematics, G2 is three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras g 2 , {\displaystyle {\mathfrak...

the Lie algebra is called solvable. The derived series for Lie algebras is analogous to the derived series for commutator subgroups in group theory, and...

Schubert variety, Schubert calculus Schur algebra, Deligne–Lusztig theory Real form (Lie theory) Weil's conjecture on Tamagawa numbers Langlands classification...

dimensions; see classification of low-dimensional real Lie algebras for up to four dimensions; and the list of Lie group topics. Column legend Cpt: Is this group...

representation theory, as first noted in the 1930s by Eugene Wigner. It links the properties of elementary particles to the structure of Lie groups and Lie algebras...

traditionally expressed in the terms that the Lie algebra of K is a real form of that of G. In the theory of algebraic groups, the relationship can also...

affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras...

orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The...

In the theory 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...

representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked out...

reductive) Lie group, can be done for all. Therefore, once the role of some low-dimensional Lie groups such as GL(2) in the theory of modular forms had been...