Lie group

English

Etymology

Named for Norwegian mathematician Sophus Lie.

Noun

Lie group (plural Lie groups)

1. (topology) Any group that is a smooth manifold and whose group operations are differentiable.
   • 1994, Silvio Levy (translator), Albert S. Schwarz, Topology for Physicists, [1989, A. S. Shvarts, Kvantovaya teoriya polya i topologiya], Springer, 1996, 2nd Printing, page 233,

     Every connected Lie group is homotopically equivalent to its maximal compact subgroup. This reduces the study of the homotopy and homology of Lie groups to the compact case.

   • 2009, Mikio Nakahara, Geometry, Topology and Physics, 2nd edition, Taylor & Francis, page 207:

     A Lie group is a manifold on which the group manipulations, product and inverse, arc defined. Lie groups play an extremely important role in the theory of fibre bundles and also find vast applications in physics.

   • 2009, Boris Khesin, Robert Wendt, The Geometry of Infinite-Dimensional Groups, Springer, page 1:

     As is well known, in finite dimensions each Lie group is, at least locally near the identity, completely described by its Lie algebra.

Hypernyms

• topological group

Hyponyms

• circle group
• Möbius group

Related terms

• group of Lie type
• Lie algebra

Translations

analytic group that is also a smooth manifold

• Danish: Liegruppe c
• Finnish: Lien ryhmä
• French: groupe de Lie (fr) m
• German: Lie-Gruppe f, Liesche Gruppe f
• Italian: gruppo di Lie m
• Japanese: リー群 (rīgun)
• Spanish: grupo de Lie m

Further reading

• Lie theory on Wikipedia.Wikipedia