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by Ciro Santilli (@cirosantilli, 37)

Lie bracket of a matrix Lie group

 ... Geometry Differential geometry Lie group Important Lie group Matrix Lie group Lie algebra of a matrix Lie group
 0 By others on same topic  0 Discussions  Updated 2025-05-07  +Created 1970-01-01  See my version
[X,Y]=XY−YX
(1)
This makes it clear how the Lie bracket can be seen as a "measure of non-commutativity"
Because the Lie bracket has to be a bilinear map, all we need to do to specify it uniquely is to specify how it acts on every pair of some basis of the Lie algebra.
Then, together with the Baker-Campbell-Hausdorff formula and the Lie group-Lie algebra correspondence, this forms an exceptionally compact description of a Lie group.

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  • Lie algebra of SL(2)

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