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Lie algebra is a group whose underlying set is smooth. For example, angles (rotations) on a circle (or units in the complex numbers) form a Lie algebra. You can

by sukilot 12y ago

Lie algebra is a group whose underlying set is smooth. For example, angles (rotations) on a circle (or units in the complex numbers) form a Lie algebra. You can add and subtract angles on a circle, and if you perturb one angle you perturb the result slightly: (theta1 + delta) + theta2 = theta1 + theta2 + delta. That's a trivial example, of course. Contrast with a symmetry group of a hexagon. There's no (non-degenerate) notion of a continuous transition from identity to rotate-by-pi.

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 ago
  Actually, 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 ago
    What do you mean by "generate" here? The exponential map is not necessarily surjective.
- alphaBetaGamma 12y ago
  Not 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.
cottonseed 12y ago
This is incorrect. A Lie _group_ is a group which is also a manifold, such that the group operations are smooth. (I don't know what it means for a set to be smooth.) A Lie _algebra_ is a vector space together with a bilinear operation (the Lie bracket, written [a, b]) which also satisfies the Jacobi identity: [a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0. How are they related? If G is a Lie group, the tangent space at the identity is a Lie algebra. What does the Lie algebra represent? Well, unlike a plain manifold, in a Lie group, given a tangent vector a, there is a path starting from a the identity in the Lie group always in the direction a. Given two tangent vectors a, b, you can flow a little in the direction a, and then a little in the direction b, or vice versa. The Lie bracket can be thought of as a measure of how the flow fails to commute.

by sukilot 12y ago

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 ago
Actually, 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 ago
  What do you mean by "generate" here? The exponential map is not necessarily surjective.
alphaBetaGamma 12y ago
Not 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.

cottonseed 12y ago

This is incorrect. A Lie _group_ is a group which is also a manifold, such that the group operations are smooth. (I don't know what it means for a set to be smooth.) A Lie _algebra_ is a vector space together with a bilinear operation (the Lie bracket, written [a, b]) which also satisfies the Jacobi identity: [a, [b, c]] + [c, [a, b]] + [b, [c, a]] = 0. How are they related? If G is a Lie group, the tangent space at the identity is a Lie algebra. What does the Lie algebra represent? Well, unlike a plain manifold, in a Lie group, given a tangent vector a, there is a path starting from a the identity in the Lie group always in the direction a. Given two tangent vectors a, b, you can flow a little in the direction a, and then a little in the direction b, or vice versa. The Lie bracket can be thought of as a measure of how the flow fails to commute.