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...

by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Let G be a Lie group, and let Ψ : G → Aut...

mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory...

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 abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital)...

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

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...

representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization...

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 the corresponding 'infinitesimal' representations of Lie algebras. A complex representation of a group is an action by a group on a finite-dimensional...

Equivalently, a representation of L is a Z2-graded representation of the universal enveloping algebra of L which respects the third equation above. A * Lie superalgebra...

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

of V. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero...

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...

{\displaystyle \pi } is unitary if and only if the associated Lie algebra representation d π : g → E n d ( H ) {\displaystyle d\pi :{\mathfrak {g}}\rightarrow...

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...

Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra g {\displaystyle {\mathfrak {g}}} over a field of...

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

and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise to an irreducible representation of the...

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...

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...

that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed...

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...

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...

representation (a spin representation). The Lie algebra g 2 {\displaystyle {\mathfrak {g}}_{2}} , being the smallest exceptional simple Lie algebra, was the first...

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

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 g0: g ≃ g...

Lie algebra of SU(2). Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation;...

In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple...

In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group...