Microworld: Lie
Groups
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above to visit the Microworld.
Authors:
Kate
Johnson and
J.T.
Ratnanather
A
Lie Group (pronounced "Lee") is a Differentiable Manifold, which satisfies
the properties of a Group, and which also has the property that its group
operations are differentiable.
A sphere and a torus are just two examples of smooth (infinitely differentiable)
manifolds. The earth, as it is a sphere, is locally "flat" and so it appears
flat when we are on it, but at a distance we can see that it is actually a
sphere. A population living on a torus would encounter a similar phenomenon.
The
torus (in the left) also has the structure of a Lie Group. The sphere (on
the right) does not. Both can be given the structure of smooth manifold, however.
Most
smooth manifolds however, are not as easy to picture and think about as are
a sphere and a torus. Although the formal definition of a smooth manifold
is complex, it will be easier to understand it in terms of a circle, sphere,
or torus, instead of as an abstract collecion of ideas.
Number
of Pages: 6
Animation: Yes
Grade Level:15-16
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