**"... A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable. ..." ( Wolfram Site )**The (self-) study of Lie Group theory is hard. I found two aids that helped me going somewhat in the subject. A book called Naive Lie Theory by John Stillwell and a series of weblectures by Erik van den Ban of the University of Utrecht. On van den Ban's homepage under Lecture Notes you'll find a Lie Group's prerequisites pdf with explanations of manifolds, tangent maps etc. which appear frequently in texts about Lie Theory.

E8 structure visualized |

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