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This article deals with Lie's theorems about Lie algebras and their associated Lie groups. For Lie's theorem on the characterization of solvable Lie algebras see Lie's theorem .

In mathematics , Lie's theorems , named after Sophus Lie , establish the connection between Lie groups and Lie algebras .

Lie groups and Lie algebras

→ Main article : Lie group

A Lie group is a differentiable manifold , which also has the structure of a group , so that the group connection and the inversion can be differentiated as often as desired . ${\displaystyle G}$

The Lie algebra of a Lie group is the vector space of the left-invariant vector fields with the commutator as Lie bracket . The Lie algebra can be canonically identified with the tangent space in the neutral element of the Lie group : ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle G}$

${\displaystyle {\mathfrak {g}}\simeq T_{e}G}$.

Lie's sentences

Theorem ( Third Lie's theorem , also Lie-Cartan 's theorem ): For every finite-dimensional real Lie algebra there is a simply connected Lie group whose Lie algebra is. ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$

Theorem ( Second Lie's theorem ): Let be Lie groups with Lie algebras and be simply connected. Then, for every Lie algebra homomorphism a unique Lie group homomorphism with . ${\displaystyle G,H}$ ${\displaystyle {\mathfrak {g}},{\mathfrak {h}}}$ ${\displaystyle G}$ ${\displaystyle f\colon {\mathfrak {g}}\to {\mathfrak {h}}}$ ${\displaystyle F\colon G\to H}$ ${\displaystyle f=D_{e}F}$

Historical and Notes

The first Lie's theorem is a purely local statement that describes the effect of a Lie group on itself in local coordinates as the solution of certain differential equations with analytic coefficients.

Even the third Lie's theorem was originally only proven in a local version by Sophus Lie ; the global form cited here goes back to Élie Cartan .

In Lie's third theorem, in addition to the simply connected Lie group, there are other (not simply connected) Lie groups with Lie algebra as a factor group , where is a discrete subgroup of the center of . ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle G/D}$ ${\displaystyle D\subset Z(G)}$ ${\displaystyle G}$

literature

• Gilmore, Robert: Lie groups, Lie algebras, and some of their applications. Reprint of the 1974 original. Robert E. Krieger Publishing Co., Inc., Malabar, FL, 1994. ISBN 0-89464-759-8
• Hilgert, Joachim; Neeb, Karl-Hermann: Structure and geometry of Lie groups. Springer Monographs in Mathematics. Springer, New York, 2012. ISBN 978-0-387-84793-1
• W. Van Est: Une demonstration de E. Cartan du troisième théorème de Lie. Actions Hamiltoniennes des groupes, troisième théorème de Lie, travail en cours, Volume 27, Hermann Paris, 1987.

Web links

• Lie's three theorems in nLab
• Robert Bryant: Cartan's generalization of Lie's third theorem (PDF; 106 kB)
• Johannes Ebert: Lie's third theorem, after Cartan-van Elst

Individual evidence

1. ^ Lie theorem Encyclopedia of Mathematics