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...

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

In mathematics, a complex Lie algebra is a Lie algebra over the complex numbers. Given a complex Lie algebra g {\displaystyle {\mathfrak {g}}} , its conjugate...

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...

This article gives a table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group...

mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups...

defined over the field of real 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...

affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra. Given...

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...

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...

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...

{\displaystyle {\mathfrak {g}}} is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), π(X) is the conjugate...

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...

designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn...

in its Lie algebra. In this case the complexification is a complex algebraic group and its Lie algebra is the complexification of the Lie algebra of the...

split real form of a complex Lie algebra, and because split semisimple Lie algebras (more generally, split reductive Lie algebras) over any field share...

of a complex Lie group. A complex semisimple Lie group is a linear algebraic group. The Lie algebra of a complex Lie group is a complex Lie algebra. A finite-dimensional...

The foundation of Lie theory is the exponential map relating Lie algebras to Lie groups which is called the Lie group–Lie algebra correspondence. The...

unital, but Lie algebras never are. The nonassociative algebra structure of A may be studied by associating it with other associative algebras which are...

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

the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because C {\displaystyle \mathbb {C} } is the Lie algebra...

associative algebra over R. Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation...

In mathematics, the complex Witt algebra, named after Ernst Witt, is the Lie algebra of meromorphic vector fields defined on the Riemann sphere that are...

algebra Lie superalgebra Abelian Lie algebra Affine Lie algebra Anyonic Lie algebra Compact Lie algebra Complex Lie algebra Exceptional Lie algebra Finite-dimensional...

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...

In mathematics, loop algebras are certain types of Lie algebras, of particular interest in theoretical physics. For a Lie algebra g {\displaystyle {\mathfrak...

In mathematics, an exceptional Lie algebra is a complex simple Lie algebra whose Dynkin diagram is of exceptional (nonclassical) type. There are exactly...

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

mathematics, the Virasoro algebra is a complex Lie algebra and the unique nontrivial central extension of the Witt algebra. It is widely used in two-dimensional...

algebras is significantly different from the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism...