CLASSIFICATION OF INVOLUTIONS OF SL(2, k)

Abstract

Abstract. In this paper we give a simple characterization of the isomorphy classes of involutions of SL(2, k) with k any field of characteristic not 2. We also classify the isomorphy classes of involutions for k algebraically closed, the real numbers, the ᒍ-adic numbers and finite fields. We determine in which cases the corresponding fixed point group H is k-anisotropic. In those cases the corresponding symmetric k-variety consists of semisimple elements.

On involutions of type O(q,k) over a field of characteristic two

“…The primary motivation is to extend Helminck's [14] study of k-involutions and symmetric k-varieties to include fields of characterstic 2. This has been studied for groups of type G 2 and A n in [19,22] and over fields of characteristic not 2 in [6,3,2,4,16,17,18]. We also extend the results of Aschbacher and Seitz [1] who studied similar structures for finite fields of characteristic 2.…”

k-Involutions of SL(n,k) over fields of characteristic 2

“…This project was inspired by the work of Wu in [9] and [6] where he determined involutions of SL(n, k) for fields of characteristic not 2, as well as for the group SO(2n + 1, k). This last was expanded on and improved by Benim and others in [3].…”

On generalized Cartan subspaces

“…Let G = SL(2, k). In [HW02] it was shown that any involution of G is isomorphic to one of the form σ a (x) = Int(A)(x) with A = 0 1 a 0 and a ∈ k * /(k * ) 2 . If H a is the fixed point group of σ a , then H a = x y ay x x 2 −ay 2 = 1 is k-anisotropic if and only if a = 1.…”