Olivetti Club

Jeffrey Jiang
Clifford algebras and spin groups

Tuesday, October 1, 2019 - 4:30pm
Malott 406

In October 1843, William Rowan Hamilton famously carved the equations $i^2 = j^2 = k^2 = ijk = -1$ into Broom Bridge, marking his discovery of the quaterinions in an act of mathematical vandalism. It was well known at the time that the complex numbers encoded a great deal of information about the geometry of the plane, since addition in $\mathbb{C}$ was vector addition, and the multiplication corresponded to scalings and rotations. Hamilton’s discovery of the quaternions constructed the analogous algebra for the geometry of $\mathbb{R}^3$. These ideas would be taken further when William Clifford constructed his eponymous algebras, which encode information of the geometry of $\mathbb{R}^n$ equipped with a nondegenerate bilinear form.

Inside the Clifford algebras, mathematicians also found a solution to a problem that had been puzzling them for some time. Certain representations of the Lie algebra $\mathfrak{so}_n$ failed to exponentiate to linear representations of $\mathrm{SO}_n$, and instead gave rise to projective representations, which stems from the fact that they arise from representations of the universal cover of $\mathrm{SO}_n$. This universal cover happens to be conveniently embedded in the group of units of a Clifford algebra, giving it an explicit description. Clifford algebras and Spin groups would prove to be extremely fruitful objects to study, and central to several important results like the Atiyah-Singer Index Theorem and Bott Periodicity.

In this talk, I will motivate the construction of Clifford algebras and Spin groups, discuss their classification, and give one characterization of Bott Periodicity.

Refreshments will be served in the lounge at 4:00 PM.