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Hooda, Neha

doi:10.1016/j.jalgebra.2014.03.039

Cited by 7 publications

(7 citation statements)

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“…A similar result holds for groups of type A 2 . We first recall, For groups of type F 4 we have the following, As a consequence of the above theorem, we have an alternative proof of ( [8], Theorem 3.4). 4 Embeddings of rank-2 tori in A 2 , G 2 and F 4…”

Section: Maximal Tori In Groups Of Type Gmentioning

confidence: 98%

“…Let H i , i = 1, 2, be algebraic subgroups of an algebraic group G. By < H 1 , H 2 > we Let G be a group of type G 2 (resp. F 4 ) defined over k. We now calculate the number of rank-2 tori required to generate G. In ( [8], Theorem 3.11, 4.1) we proved that a group of type G 2 is generated by its k-subgroups of type A 1 and a group of type F 4 is generated by its k-subgroups of type A 2 . The results below are continuation of that.…”

Section: The Real Casementioning

confidence: 99%

“…The main aim of this paper is to investigate embeddings of rank-2 tori in groups of type G 2 , F 4 and A 2 . In our earlier work ( [8]), we studied k-embeddings of connected, simple algebraic groups of type A 1 and A 2 in simple groups of type G 2 and F 4 , defined over k, in terms of their respective mod-2 Galois cohomological invariants. We also showed that these groups are generated by their A 2 type k-subgroups.…”

Section: Introductionmentioning

confidence: 99%

“…3.3. ([8], Prop.3.4) Let F be a quadratic étale k-algebra and B be a degree 3 central simple algebra over F with an involution σ of the second kind. Then σ is distinguished over k if and only if SU(B, σ) becomes isotropic over an odd degree extension.…”

mentioning

confidence: 99%

“…Let H =< H 1 , H 2 >. By Lemma 6.1, H is a connected, reductive, k-anisotropic, nontoral subgroup of G. By ([8], Theorem 3.10), [H, H] is of type A 2 , A 2 × A 2 , D 4 or F 4 . Since D 4 A 2 , A 2 × A 2 , [H, H] is of type D 4 or F 4 .…”

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Embeddings of rank-2 tori in algebraic groups

Hooda

2018

Journal of Pure and Applied Algebra

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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 * .

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Section: Maximal Tori In Groups Of Type Gmentioning

confidence: 98%

Section: The Real Casementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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