Familiarity with linear algebra, including matrix mutiplication, change of basis, determinant, trace, eigenvalues and eigenvectors;
Some exposure to abstract algebra, including vector spaces and groups;
Comfort with differential calculus;
Some acquaintance with complex numbers.
Course description
This course will discuss the geometry of both Lie groups and Lie algebras, with an emphasis on the exceptional cases and their description in terms of the octonions. Lie groups are groups of continuous symmetries, generalizing the familiar notion of rotation groups; Lie algebras are their infinitesimal versions. Symmetry groups describe many physical phenomena, and Lie groups are widely used in physics, notably in the description of fundamental particles.
In the late 1800s, Killing and Cartan classified the simple Lie algebras into 4 infinite classical families and 5 exceptional cases. The classical Lie algebras correspond to matrix groups over the reals, the complexes, and Hamilton's quaternions. In the 1960s, a unified description of the exceptional Lie algebras was given by Freudenthal and Tits in terms of the octonions, the largest of the four division algebras.
The goal of this course is to describe the structure of the corresponding exceptional Lie groups, utilizing tools from both geometry and algebra. We will discuss the general structure of Lie algebras and Lie groups, the classification theorem, and the Freudenthal–Tits magic square of Lie algebras, culminating in a treatment of the largest of the exceptional Lie groups, $E_8$, in terms of the octonions.