ag.algebraic geometry - Learning about Lie groups

Nobody mentioned "Gilmore: Lie Groups, Physics, and Geometry" yet.

A very down to earth introduction with many examples and clear explanations. Especially targeted at physicists, engineers and chemists.

If you follow the above link you can read some sample chapters.

The cover summarizes the set up of the book quite neatly:

"Describing many of the most important aspects of Lie group theory, this book
presents the subject in a ‘hands on’ way. Rather than concentrating on theorems
and proofs, the book shows the relation of Lie groups with many branches of
mathematics and physics, and illustrates these with concrete computations. Many
examples of Lie groups and Lie algebras are given throughout the text, with applications
of the material to physical sciences and applied mathematics. The relation
between Lie group theory and algorithms for solving ordinary differential equations
is presented and shown to be analogous to the relation between Galois groups
and algorithms for solving polynomial equations. Other chapters are devoted to
differential geometry, relativity, electrodynamics, and the hydrogen atom.
Problems are given at the end of each chapter so readers can monitor their
understanding of the materials. This is a fascinating introduction to Lie groups
for graduate and undergraduate students in physics, mathematics and electrical
engineering, as well as researchers in these fields.

Robert Gilmore is a Professor in the Department of Physics at Drexel University,
Philadelphia. He is a Fellow of the American Physical Society, and a Member
of the Standing Committee for the International Colloquium on Group Theoretical
Methods in Physics. His research areas include group theory, catastrophe theory,
atomic and nuclear physics, singularity theory, and chaos."