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Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

28 votes
Accepted

Origin of terms "flag", "flag manifold", "flag variety"?

Armand Borel's Bourbaki Seminar 121 Groupes algébriques is from 1955, and uses "drapeau" (page 7). (It's online at archive.numdam.org.) This may not be the earliest occurrence, but there is a good rea …
j.c.'s user avatar
  • 13.6k
14 votes

What is the situation with Hilbert's Fifth Problem?

http://en.wikipedia.org/wiki/Hilbert%27s_fifth_problem is a decent survey. In general in the discussion of "status" of the Hilbert problems, there are at least two recognisable routes. Route A is the …
Martin Sleziak's user avatar
7 votes

Why study Lie algebras?

As has been said, Lie groups are our best theory encoding continuous symmetry. Lie algebra theory, which is the infinitesimal counterpart, is a theory good enough that numerous problems can be solved …
Charles Matthews's user avatar
1 vote

Parametrization of O(3)

If there were a really simple way we wouldn't need the concept of "gimbal lock" (http://en.wikipedia.org/wiki/Gimbal_lock). In other words the manifold in question is compact but isn't the 3-torus, an …
Charles Matthews's user avatar
3 votes
Accepted

Is the object we get when we quotient $U(N)$ by $U(N-k)$ familar?

Putting a name to the space, it's a complex Stiefel manifold. See http://en.wikipedia.org/wiki/Stiefel_manifold. (But I wasn't the first.)
Charles Matthews's user avatar