Statements (54)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:class |
classified by maximal torus and root system
|
| gptkbp:example |
gptkb:special_orthogonal_group_SO(n)
gptkb:symplectic_group_Sp(n) gptkb:circle_group_S^1 gptkb:orthogonal_group_O(n) gptkb:special_unitary_group_SU(n) gptkb:torus_T^n gptkb:unitary_group_U(n) |
| https://www.w3.org/2000/01/rdf-schema#label |
compact Lie groups
|
| gptkbp:property |
gptkb:Riemannian_manifold
finite fundamental group for simple compact Lie groups every compact Lie group is complete as a metric space every compact Lie group is a closed subgroup of U(n) for some n all closed subgroups are Lie subgroups compact topology connected component is a Lie group every compact Lie group has a Cartan subgroup every compact Lie group has a Weyl group every compact Lie group has a bi-invariant metric every compact Lie group has a maximal torus every compact Lie group has a root system every compact Lie group is Hausdorff every compact Lie group is a real Lie group every compact Lie group is a real algebraic group every compact Lie group is locally path-connected every compact Lie group is metrizable every compact Lie group is second countable every compact Lie group is unimodular every representation is completely reducible finite center for simple compact Lie groups finite-dimensional representations group structure has Haar measure every compact Lie group is a finite extension of a torus by a semisimple group every compact Lie group is a real analytic manifold every compact Lie group is isomorphic to a matrix group every compact Lie group is a projective limit of Lie groups all irreducible representations are finite-dimensional |
| gptkbp:relatedTo |
gptkb:Weyl_group
gptkb:Peter–Weyl_theorem gptkb:homogeneous_spaces gptkb:Lie_group representation theory of Lie groups |
| gptkbp:studiedIn |
differential geometry
mathematical physics representation theory |
| gptkbp:subclassOf |
gptkb:topology
gptkb:Lie_group |
| gptkbp:bfsParent |
gptkb:Weyl's_theorem
gptkb:Weyl_character_formula gptkb:Lie_group gptkb:Lie_groups |
| gptkbp:bfsLayer |
5
|