Compact Lie algebra information

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...

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...

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...

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

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...

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

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...

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

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...

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

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...

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...

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

In mathematics, the special linear Lie algebra of order n over a field F {\displaystyle F} , denoted s l n F {\displaystyle {\mathfrak {sl}}_{n}F} or...

Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group...

semisimple Lie algebra or abelian Lie algebra is a fortiori reductive. Over the real numbers, compact Lie algebras are reductive. A Lie algebra g {\displaystyle...

enveloping algebra Baker-Campbell-Hausdorff formula Casimir invariant Killing form Kac–Moody algebra Affine Lie algebra Loop algebra Graded Lie algebra One-parameter...