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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The one parameter subgroup of a Lie group for a given element of its Lie algebra is a subgroup of given by:
Intuitively, is a direction, and is how far we move along a given direction. This intuition is especially vivid in for example in the case of the Lie algebra of , the rotation group.
One parameter subgroups can be seen as the continuous analogue to the cycle of an element of a group.
You can learn more from older students than from faculty Updated 2025-07-14 +Created 1970-01-01
Wikipedia mentions quoting his Nobel Prize biography:
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The Questioning City by British Pathé (1963)
Source. Located in Cambridge, United Kingdom.
Second largest number of Nobel Prize as of 2019, need on say more? en.wikipedia.org/w/index.php?title=List_of_Nobel_laureates_by_university_affiliation&oldid=934987075
Lie Groups, Physics, and Geometry by Robert Gilmore (2008) 7.2 "The covering problem" gives some amazing intuition on the subject as usual.
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