Speaker: Miles A. Reid FRS (University of Warwick)

Time: 16:00-17:00 p.m., May 27, 2025, GMT+8

Venue: Room 77201, Jingchunyuan 78, BICMR, PKU

Abstract: 

The family denoted D_n is a main trunk of the classification of simple algebraic groups. For n >= 5, D_n corresponds to SO(2n) and its double cover Spin(2n) -> SO(2n). The spin groups are traditionally constructed via Clifford algebras. It is instructive to follow this trunk down to its roots. In particular, the case D_2 is the tiny acorn from which that whole structure grow.

Of course the Dynkin diagram D_2 is not usually allowed, as the union of two disjoint copies of A_1. I use completely elementary arguments based on the quadric surface Q : (x*y=z*t) in PP^3 and its two family of lines (that make Q isomorphic to PP^1 x PP^1) to display Spin(4) as isomorphic to SL(2) x SL(2).

In the same vein SO(8) corresponds to the Dynkin diagram D_4 that should be drawn with S3 symmetry. The double cover Spin(8) -> SO(8) displays the unique phenomenon called triality. This is usually discussed in terms of the algebra of Octonions (or the Cayley numbers), but I construct the group Spin(8) and its representation theory from simpler arguments using projective homogeneous spaces.

Source: Beijing International Center for Mathematical Research, PKU