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Ah! So its just a group that has a defined group of continuous transformations that can be applied to it?
by aswanson 12y ago
Ah! So its just a group that has a defined group of continuous transformations that can be applied to it?
- dougabug 12y agoActually, a Lie Group is a group which is simultaneously a manifold, such that the group multiplication corresponds to diffeomorphisms of the manifold onto itself. The Lie Algebra corresponding to that Lie Group is the tangent space of the identity. The elements of the Lie Algebra generate the given Lie Group via the exponential map.
- cottonseed 12y agoWhat do you mean by "generate" here? The exponential map is not necessarily surjective.
- alphaBetaGamma 12y agoNot really. Very roughly: It's a group with a an infinite number of elements which is smooth, in the sense that there is a notion that elements can be close or far from each other, and locally around each element the group looks like R^n (for some n). Of course the group operation must play nice with this notion of "close". I.e. if elements A and B are close, after multiplying both of them by X, AX and BX can't be too far: the group operation does not "tear" the structure apart. So in the example above of rotations on the plane, you can visualize the group as a circle where a point represents the rotation by that angle. Locally it looks like R, but it has a different global structure.