semisimple Lie groups
E503516
Semisimple Lie groups are a class of Lie groups whose Lie algebras decompose into simple components and play a central role in representation theory, geometry, and mathematical physics.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| semisimple Lie group | 0 |
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
Lie group
ⓘ
mathematical concept ⓘ |
| characterizedBy |
Killing form of its Lie algebra is nondegenerate
ⓘ
radical of its Lie algebra is zero ⓘ |
| classifiedBy |
Cartan matrix
NERFINISHED
ⓘ
Dynkin diagram NERFINISHED ⓘ root system ⓘ |
| definedBy | its Lie algebra is a direct sum of simple Lie algebras ⓘ |
| hasExample |
exceptional Lie group E6
NERFINISHED
ⓘ
exceptional Lie group E7 ⓘ exceptional Lie group E8 NERFINISHED ⓘ exceptional Lie group F4 ⓘ exceptional Lie group G2 NERFINISHED ⓘ special linear group SL(n,ℂ) NERFINISHED ⓘ special linear group SL(n,ℝ) NERFINISHED ⓘ special orthogonal group SO(n) NERFINISHED ⓘ symplectic group Sp(2n,ℝ) NERFINISHED ⓘ |
| hasPart | simple Lie group ⓘ |
| hasProperty |
Lie algebra is semisimple
ⓘ
connected (often assumed in structure theory) ⓘ finite center (often assumed in classification) ⓘ no nontrivial connected solvable normal subgroup ⓘ |
| hasRepresentationTheoryProperty |
finite-dimensional representations are completely reducible
ⓘ
unitary dual is central object of harmonic analysis ⓘ |
| hasStructure |
Cartan subgroup
ⓘ
Weyl group NERFINISHED ⓘ root space decomposition ⓘ |
| playsRoleIn |
algebraic geometry
ⓘ
automorphic forms ⓘ differential geometry ⓘ harmonic analysis ⓘ mathematical physics ⓘ number theory ⓘ particle physics ⓘ quantum field theory ⓘ representation theory ⓘ |
| relatedConcept |
algebraic group
ⓘ
compact Lie group ⓘ reductive Lie group NERFINISHED ⓘ semisimple Lie algebra ⓘ |
| studiedUsing |
Bruhat decomposition
NERFINISHED
ⓘ
Cartan decomposition NERFINISHED ⓘ Iwasawa decomposition NERFINISHED ⓘ maximal compact subgroup ⓘ |
| subclassOf |
complex Lie group
ⓘ
real Lie group ⓘ |
| usedIn |
classification of elementary particles via symmetry groups
ⓘ
gauge theories in physics ⓘ theory of symmetric spaces ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.