About: Abelian Lie group

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In geometry, an abelian Lie group is a Lie group that is an abelian group. A connected abelian real Lie group is isomorphic to . In particular, a connected abelian (real) compact Lie group is a torus; i.e., a Lie group isomorphic to . A connected complex Lie group that is a compact group is abelian and a connected compact complex Lie group is a complex torus; i.e., a quotient of by a lattice.

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  • In geometry, an abelian Lie group is a Lie group that is an abelian group. A connected abelian real Lie group is isomorphic to . In particular, a connected abelian (real) compact Lie group is a torus; i.e., a Lie group isomorphic to . A connected complex Lie group that is a compact group is abelian and a connected compact complex Lie group is a complex torus; i.e., a quotient of by a lattice. Let A be a compact abelian Lie group with the identity component . If is a cyclic group, then is topologically cyclic; i.e., has an element that generates a dense subgroup. (In particular, a torus is topologically cyclic.) (en)
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  • In geometry, an abelian Lie group is a Lie group that is an abelian group. A connected abelian real Lie group is isomorphic to . In particular, a connected abelian (real) compact Lie group is a torus; i.e., a Lie group isomorphic to . A connected complex Lie group that is a compact group is abelian and a connected compact complex Lie group is a complex torus; i.e., a quotient of by a lattice. (en)
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  • Abelian Lie group (en)
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