Embeddings of rank-2 tori in algebraic groups

Abstract

Let k be a field of characteristic different from 2 and 3. In this paper we study connected simple algebraic groups of type A 2 , G 2 and F 4 defined over k, via their rank-2 k-tori. Simple, simply connected groups of type A 2 play a pivotal role in the study of exceptional groups and this aspect is brought out by the results in this paper. We refer to tori, which are maximal tori of A n type groups, as unitary tori. We discuss conditions necessary for a rank-2 unitary k-torus to embed in simple k-groups of type A 2 , G 2 and F 4 in terms of the mod-2 Galois cohomological invariants attached with these groups. We calculate the number of rank-2 k-unitary tori generating these algebraic groups (in fact exhibit such tori explicitly). The results in this paper and our earlier paper ([8]) show that the mod-2 invariants of groups of type G 2 , F 4 and A 2 are controlled by their k-subgroups of type A 1 and A 2 as well as the unitary k-tori embedded in them.(1) L/k (G m ) we shall denote the k-torus Ker{N L/k : L * → k * } of norm 1 elements in L and, when convenient, this torus will also be denoted by L (1) . Let G be an algebraic group defined over k. By G(k) we denote the group of k-rational points in G. For a finiteétale extension L of k the group of k-points of L (1) will be denoted by L (1) . We define an algebraic group G to be simple if G has no non-trivial proper connected normal subgroups. By << a 1 , a 2 , · · · , a n >> we shall mean the n-fold Pfister form < 1, −a 1 > ⊗ < 1, −a 2 > ⊗ · · · ⊗ < 1, −a n > over k. In the paper, unadorned tensor products will be understood to be over base fields and dimensions, when not specified, are over the base fields. For an object X defined over k, X ⊗ k L will denote the base change X × k L of X . By K = k( √ α) we denote the quadratic algebra k[x]/(x 2 − α) for α ∈ k * .

Embeddings of maximal tori in groups of type $F_4$

“…We turn to the situation for exceptional groups. If G is of type G 2 , embeddings of maximal tori in G have been studied in [BGL15] and [Hoo18]. For outer forms of type D 4 (in particular, triality forms), classifications and local-global principles in many cases have been established in [Fio17].…”