Statements (24)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:defines |
Lie groups that are connected as topological spaces
|
| gptkbp:example |
gptkb:special_linear_group_SL(n,_R)
gptkb:unitary_group_U(n) gptkb:general_linear_group_GL^+(n,_R) gptkb:orthogonal_group_SO(n) |
| https://www.w3.org/2000/01/rdf-schema#label |
connected Lie groups
|
| gptkbp:property |
every connected Lie group is a homogeneous space under itself
the fundamental group of a connected Lie group is abelian the exponential map is locally surjective for connected Lie groups connected Lie groups can be non-compact or compact every connected Lie group is path-connected the universal cover of a connected Lie group is a Lie group the identity component of a Lie group is a connected Lie group connected Lie groups can be simple, semisimple, solvable, or nilpotent any Lie group has a unique maximal connected subgroup containing the identity connected Lie groups are locally isomorphic if their Lie algebras are isomorphic connected Lie groups are classified by their Lie algebras and fundamental groups |
| gptkbp:studiedIn |
gptkb:Lie_theory
differential geometry |
| gptkbp:subclassOf |
gptkb:Lie_group
|
| gptkbp:bfsParent |
gptkb:Lie_group
gptkb:Lie_groups |
| gptkbp:bfsLayer |
5
|