Constructions with regular degenerate quadratic forms

Abstract

This chapter uses degenerate quadratic forms and quadrics in Severi–Brauer variety to give a geometric description of all non-standard absolutely pseudo-simple k-groups G of minimal type with root system Bn over ks such that ZG = 1 and the Cartan k-subgroups of G are tori. It begins with an overview of the lemma and propositions for regular degenerate quadratic forms, coupled with two examples. It then considers the conformal isometry between quadratic spaces over a field, which is a linear isomorphism that respects the quadratic forms up to a nonzero scaling factor. It also introduces a proposition that provides sufficient conditions for an absolutely pseudo-simple k-group to be isomorphic to SO(q) for a regular quadratic form q. Finally, it describes all descents in terms of automorphisms of certain quadrics in Severi–Brauer varieties over k.