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 s

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.