The work of Norwegian mathematician Sophus Lie extends ideas of symmetry and leads to many applications in mathematics and physics. Ordinarily, the study of the "objects" in Lie's theory (Lie groups and Lie algebras) requires extensive mathematical prerequisites beyond the reach of the typical undergraduate. By restricting to the special case of matrix Lie groups and relying on ideas from multivariable calculus and linear algebra, this lovely and important material becomes accessible even to college sophomores. Working with Lie's ideas fosters an appreciation of the unity of mathematics and the sometimes surprising ways in which mathematics provides a language to describe and understand the physical world. This is the only book in the undergraduate curriculum to bring this material to students so early in their mathematical careers.
1. Symmetries of vector spaces
2. Complex numbers, quaternions and geometry
4. One-parameter subgroups and the exponential map
5. Lie algebras
6. Matrix groups over other fields
Appendix I. Linear algebra facts
Appendix II. Paper assignment used at Mount Holyoke College
Appendix III. Opportunities for further study
Solutions to selected problems
Harriet Pollatsek (Mount Holyoke College, South Hadley, MA) has served as chair of the MAA's Committee on the Undergraduate Program in Mathematics and led the writing team for Undergraduate Programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide 2004. She now chairs the MAA's Council on Programs and Students in the Mathematical Sciences.