Lie Groups

1. Symmetries of vector spaces: 1.1. What is a symmetry?; 1.2. Distance is fundamental; 1.3. Groups of symmetries; 1.4. Bilinear forms and symmetries of spacetime; 1.5. Putting the pieces together; 1.6. A broader view: Lie groups; 2. Complex numbers, quaternions and geometry: 2.1. Complex numbers; 2.2. Quaternions; 2.3. The geometry of rotations of R3; 2.4. Putting the pieces together; 2.5. A broader view: octonions; 3. Linearization: 3.1. Tangent spaces; 3.2. Group homomorphisms; 3.3. Differentials; 3.4. Putting the pieces together; 3.5. A broader view: Hilbert's fifth problem; 4. One-parameter subgroups and the exponential map: 4.1. One-parameter subgroups; 4.2. The exponential map in dimension one; 4.3. Calculating the matrix exponential; 4.4. Properties of the matrix exponential; 4.5. Using exp to determine L(G); 4.6. Differential equations; 4.7. Putting the pieces together; 4.8. A broader view: Lie and differential equations; 4.9. Appendix on convergence; 5. Lie algebras: 5.1. Lie algebras; 5.2. Adjoint maps { big `A' and small `a'; 5.3. Putting the pieces together; 5.4. A broader view: Lie theory; 6. Matrix groups over other fields: 6.1. What is a field?; 6.2. The unitary group; 6.3. Matrix groups over finite fields; 6.4. Putting the pieces together; 6.5. A broader view of finite groups of Lie type and simple groups; Appendix I. Linear algebra facts; Appendix II. Paper assignment used at Mount Holyoke College; Appendix III. Opportunities for further study; Solutions to selected problems; Bibliography.