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 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...

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

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

Lie groups whose Lie algebras are semisimple Lie algebras. The Lie algebra of a simple Lie group is a simple Lie algebra. This is a one-to-one correspondence...

algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of simple Lie algebras is called a semisimple Lie algebra....

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

notions of complexification and real form have a natural description in the language of algebraic geometry. Just as complex semisimple Lie algebras are classified...

finite-dimensional representations of Lie algebras Weights in the representation theory of semisimple Lie algebras Hall 2015 Section 1.2 Bröcker & tom...

theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation theory...

develop the theory of complex semisimple Lie algebras by viewing them as the complexifications of Lie algebras of compact groups; the existence of an Ad-invariant...

articles on representation theory of a connected compact Lie group and the parallel theory classifying representations of semisimple Lie algebras. Let T be...

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...

classification and representation theory of semisimple Lie algebras. Since Lie groups (and some analogues such as algebraic groups) and Lie algebras have become...

Lie algebras are analogs of solvable groups. Any nilpotent Lie algebra is a fortiori solvable but the converse is not true. The solvable Lie algebras...

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

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

Kac–Moody algebras and generalized Kac–Moody algebras also have subalgebras that play the same role as the Cartan subalgebras of semisimple Lie algebras (over...

many properties with semisimple Lie algebras over algebraically closed fields – having essentially the same representation theory, for instance – the splitting...

nilpotent Lie algebras are analogs of nilpotent groups. The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras, by successive...

representation of semisimple Lie algebras that plays such a fundamental role in present-day Lie theory. And although Lie envisioned applications of his...

points of view; see the representation theory of semisimple Lie algebras and the Weyl character formula. The functor that takes an associative algebra A over...

modular representation theory, and more generally in the theory of representations of finite-dimensional associative algebras A that are not semisimple, a...

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

particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system...

The representation theory of SU(3) is well-understood. Descriptions of these representations, from the point of view of its complexified Lie algebra s l...

representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra...

of the existence of mass and spin (normally explained in Wigner's classification of relativistic mechanics) in terms of the representation theory of the...

representations of Lie algebras Representation theory of a connected compact Lie group Weights in the representation theory of semisimple Lie algebras Dixmier...

the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups...