About: Lie group–Lie algebra correspondence

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In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvious counterexample is and (see real coordinate space and the circle group respectively) which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other. However, by restricting our attention to the simply connected Lie groups, the Lie group-Lie algebra correspondence will be one-to-one.

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rdf:type	planet
rdfs:label	Correspondencia grupo de Lie-álgebra de Lie (es) Lie group–Lie algebra correspondence (en) 리 대응 (ko)
rdfs:comment	En matemáticas, la correspondencia entre el grupo de Lie y el álgebra de Lie permite estudiar los grupos de Lie, que son objetos geométricos, en términos de álgebras de Lie, que son objetos lineales. En este artículo, cuando se habla de un grupo de Lie se hace referencia a un grupo de Lie real. Para los casos complejos y p-ádicos, véase el grupo de Lie complejo y el grupo de Lie p-ádico. En este artículo, se supone que las variedades (en particular, los grupos de Lie) son los segundos numerables; en particular, tienen como máximo varios componentes conectados. (es) 리 군론에서 리 대응(Lie對應, 영어: Lie correspondence)은 리 군의 범주에서 실수 리 대수의 범주로 가는 표준적인 함자이다. 즉, 각 리 군에 표준적 실수 리 대수가 대응되며, 리 군의 매끄러운 군 준동형에 실수 리 대수의 준동형이 대응된다. (ko) In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvious counterexample is and (see real coordinate space and the circle group respectively) which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other. However, by restricting our attention to the simply connected Lie groups, the Lie group-Lie algebra correspondence will be one-to-one. (en)
dcterms:subject	Differential geometry Lie algebras Lie groups Manifolds
Wikipage page ID	43302095 (xsd:integer)
Wikipage revision ID	1111909962 (xsd:integer)
Link from a Wikipage to another Wikipage	De Rham cohomology Vector field Lie's third theorem Lie algebra cohomology Lie algebra representation Lie group Differential geometry Lie algebras Complexification Mathematics SO(n) SU(n) General linear group Full and faithful functor Milnor–Moore theorem Representation theory of SU(2) Lie algebra So(n) Submersion (mathematics) Compact Lie algebra Complex Lie group Complex torus Functor Fundamental group Closed subgroup theorem Hopf algebra Malcev Lie algebra Tangent space Baker–Campbell–Hausdorff formula Ado's theorem Central extension (mathematics) Distribution on a linear algebraic group Adjoint functor Adjoint representation Lie groups Manifolds Differential (mathematics) Isomorphism Simply connected space Universal enveloping algebra First isomorphism theorem Adjoint representation of a Lie algebra Chain rule Killing form Ehresmann's lemma Rotation group SO(3) Diagonalizable matrix Direct product of groups Distribution (mathematics) Sp(n) Special unitary group Spin (physics) Circle group Group law Category (mathematics) Category theory Real coordinate space Lie bracket of vector fields Root system Unimodular group Exponential map (Lie theory) Immersion (mathematics) Formal group Principal bundle P-adic Lie group Lie algebra homomorphism Primitive element (Hopf algebra) Second countable Finite-dimensional representation Lie group homomorphism One-parameter subgroup One to one correspondence Classical compact groups Sl(n) Adjoint group Left invariant Simply connected covering dbr:Integer_lattice_(Lie_group) dbr:Rigidity_argument
Link from a Wikipage to an external page	http://amathew.wordpress.com/2012/06/23/formal-lie-theory-in-characteristic-zero/ http://people.brandeis.edu/~syzygy/LieGroups.pdf
sameAs	Lie group–Lie algebra correspondence Lie group–Lie algebra correspondence Lie group–Lie algebra correspondence Lie group–Lie algebra correspondence Lie group–Lie algebra correspondence Lie group–Lie algebra correspondence

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