Topological groups, Lie groups Structure of Lie groups Concept
Semisimple Lie group
Summary
A semisimple Lie group is a connected Lie group whose Lie algebra is a semisimple Lie algebra, that is, has no nonzero solvable ideals. Equivalently, its Lie algebra is a direct sum of simple Lie algebras. So, up to a covering and a central quotient, the group is a product of simple Lie groups. Semisimple Lie groups have a rich structure theory, with Cartan, Iwasawa and Bruhat decompositions, and a deep representation theory. By the Levi decomposition, a finite-dimensional real or complex Lie algebra is a semisimple subalgebra plus its largest solvable ideal. So semisimple and solvable groups are the two basic kinds.
Where this leads
20 pages build on this one.
- Iwasawa decomposition of a Lie group
- Cartan–Iwasawa–Malcev theorem
- Principal series representation
- Cartan decomposition of a Lie group
- Margulis normal subgroup theorem
- Langlands decomposition
- Kazhdan–Margulis theorem
- KAK decomposition
- Levi–Malcev decomposition
- Margulis arithmeticity theorem
- Flag manifold
- Borel–Weil–Bott theorem
- Parabolic subgroup
- Bruhat decomposition of a Lie group
- Wang's finiteness theorem
- Zimmer program
- Borel density theorem
- Weil's local rigidity theorem
- Calabi–Weil rigidity
- Langlands classification
Further reading
About this page
| Tag | XJS07AJ50Y |
|---|---|
| URI | m2://XJS07AJ50Y |
| Kind | concept |
| Writing | W0 stub |
| MSC | 22E46, 22E15, 22E10 |
| Specializes | Reductive Lie group |
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Definitions
Results about this
- Iwasawa decomposition of a Lie group
- Cartan decomposition of a Lie group
- Kazhdan–Margulis theorem
- KAK decomposition
- Bruhat decomposition of a Lie group
Used by
- Iwasawa decomposition of a Lie group
- Cartan decomposition of a Lie group
- Margulis normal subgroup theorem
- Langlands decomposition
- Kazhdan–Margulis theorem
- KAK decomposition
- Levi–Malcev decomposition
- Margulis arithmeticity theorem
- Flag manifold
- Borel–Weil–Bott theorem
- Parabolic subgroup
- Bruhat decomposition of a Lie group
- Wang's finiteness theorem
- Zimmer program
- Borel density theorem
- Weil's local rigidity theorem