Master Degree Project Presentation: Subgroups of algebraic groups of type G2 via octonion algebras

Abstract: Over an algebraically closed field, every octonion algebra is isomorphic to the split octonion algebra, which can be seen as pairs of 2x2-matrices, (x,y), with norm q(x,y)=det(x)-det(y). The exceptional algebraic group of type G2 can be realized as the automorphism group of an octonion algebra.

Using this, we express some subgroups of G2 – including its maximal tori, root subgroups, Borel subgroups, and parabolic subgroups – in terms of octonions. For maximal tori, we extend a result by T.A. Springer and F. D. Veldkamp. For root subgroups, we use the explicit action via the root homomorphisms on GL7, building on computations by M. Maccan. For the remaining subgroups, we use their relation to maximal tori and Borel subgroups.