field of Lie theory, there are two definitions of a compact Lie algebra. Extrinsically and topologically, a compact Lie algebra is the Lie algebra of a compact...

used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, R {\displaystyle...

a compact Lie algebra. It is known that under the Lie correspondence, compact Lie algebras correspond to compact Lie groups. The compact form corresponds...

semidirect sum of a central extension of the super-Poincaré algebra by a compact Lie algebra B of internal symmetries. Bosonic fields commute while fermionic...

mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero...

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 (or endomorphisms...

In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras...

mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an...

classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note...

representations of a simply connected compact Lie group K and the finite-dimensional representations of the complex semisimple Lie algebra g {\displaystyle {\mathfrak...

semisimple Lie algebras, split Lie algebras are opposite to compact Lie algebras – the corresponding Lie group is "as far as possible" from being compact. The...

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

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

representation theory of a connected compact Lie group and the parallel theory classifying representations of semisimple Lie algebras. Let T be a maximal torus in...

maximal torus of the compact group. If g {\displaystyle {\mathfrak {g}}} is a linear Lie algebra (a Lie subalgebra of the Lie algebra of endomorphisms of...

symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of...

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...

objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is 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...

Lie algebra g {\displaystyle {\mathfrak {g}}} is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie...

compact group and X is a skew-adjoint operator in its Lie algebra. In this case the complexification is a complex algebraic group and its Lie algebra...

semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using...

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

supersymmetry algebra, defined as the semidirect product of a central extension of the super-Poincaré algebra by a compact Lie algebra of internal symmetries...

In mathematics, a Lie algebra g {\displaystyle {\mathfrak {g}}} is nilpotent if its lower central series terminates in the zero subalgebra. The lower...

a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra...

constants forms a Lie algebra under the Poisson bracket. This Lie algebra is a one-dimensional central extension of the commutative Lie algebra R 2 n {\displaystyle...

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...

F4 is a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is...

positive weight. A Lie subalgebra of g {\displaystyle {\mathfrak {g}}} containing a Borel subalgebra is called a parabolic Lie algebra. Hyperbolic group...