Hello! I've read some lie algebra in both group theory books and differential geometry books, and is confused about the different perspectives. The group theory approach is usually that the author introduce the generators and show that these fulfil the algebra. In differential geometry our lie group is a differential manifold for which the algebra is found by left invariant fields or the tangent space.. I.e. in group theory our algebra is fulfilled by matrices and in diff. geometry use the differential operators that span the tangentspace. Are we talking about the same algebra? How do they relate? Does the generators describe the partial derivatives/ our basis in T_eM? Etc. :) Please explain! Thanks.