Involutions of Type G 2 Over Fields of Characteristic Two
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Involutions of Type G2 Over Fields of Characteristic Two John Hutchens1
· Nathaniel Schwartz2
Received: 30 November 2016 / Accepted: 6 July 2017 © Springer Science+Business Media B.V. 2017
Abstract We continue a study of automorphisms of order 2 of algebraic groups. In particular we look at groups of type G2 over fields k of characteristic two. Let C be an octonion algebra over k; then Aut(C) is a group of type G2 over k. We characterize automorphisms of order 2 and their corresponding fixed point groups for Aut(C) by establishing a connection between the structure of certain four dimensional subalgebras of C and the elements in Aut(C) that induce inner automorphisms of order 2. These automorphisms relate to certain quadratic forms which, in turn, determine the Galois cohomology of the fixed point groups of the involutions. The characteristic two case is unique because of the existence of four dimensional totally singular subalgebras. Over finite fields we show how our results coincide with known results, and we establish a classification of automorphisms of order 2 over infinite fields of characteristic two. Keywords Algebraic groups · Symmetric spaces · Octonions · Involutions · Exceptional groups · Symmetric k-varieties · Quadratic forms Mathematics Subject Classification (2010) 20D45 · 20E36 · 20G41
Presented by Jon F. Carlson. John Hutchens
[email protected] Nathaniel Schwartz [email protected] 1
Winston-Salem State University, 601 S. Martin Luther King Dr., Winston-Salem, NC 27110, USA
2
Washington College, 300 Washington Ave., Chestertown, MD 21620, USA
J. Hutchens, N. Schwartz
1 Introduction The study of Riemannian symmetric spaces has provided important tools for extending the representation theory of Lie groups, as well as a variety of other connections in mathematics and physics. Cartan introduced symmetric spaces of real Lie groups in 1926–27 in his seminal papers [5, 6] where he also gave a complete classification. His results were extended by Gantmacher [11] and others, and in 1957 Berger [3] classified symmetric spaces of real Lie algebras. These and other works formed a basis that Helminck extended in [12] to what he calls symmetric k-varieties (or generalized symmetric spaces) of linear algebraic groups defined over arbitrary fields k. Symmetric k-varieties provide functionally similar tools to symmetric spaces, yielding results over the broader class of linear algebraic groups defined over arbitrary fields. Helminck also shows that symmetric k-varieties of algebraic groups and isomorphism classes of k-involutions form a one-to-one correspondence when the characteristic is not two. Helminck gave a classification scheme in [13] of symmetric k-varieties over fields of characteristic not two, the main ingredients of which are automorphisms of order 2 together with their fixed point groups. The construction proceeds as follows. We take an algebraic group G and some k-involution θ ∈ Aut(G), such that θ 2 = id and θ(G(k)) = G(k), the group of k-rational points of G. We use the fixed
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