Isomorphy classes of k-involutions of G2

Abstract

Isomorphy classes of k-involutions have been studied for their correspondence with symmetric k-varieties, also called generalized symmetric spaces. A symmetric k-variety of a k-group G is defined as Gk/Hk where θ : G → G is an automorphism of order 2 that is defined over k and Gk and Hk are the k-rational points of G and H = Gθ, the fixed point group of θ, respectively. This is a continuation of papers written by A. G. Helminck and collaborators [Involutions of SL (2, k), (n > 2), Acta Appl. Math.90(1–2) (2006) 91–119, Classification of involutions of SO (n; k; b), to appear, On the classification of k-involutions, Adv. Math.153(1) (1988) 1–117, Classification of involutions of SL (2, k), Comm. Algebra30(1) (2002) 193–203] expanding on his combinatorial classification over certain fields. Results have been achieved for groups of type A, B and D. Here we begin a series of papers doing the same for algebraic groups of exceptional type.