Geometries, the principle of duality, and algebraic groups

Garibaldi, Skip; Carr, Michael

doi:10.1016/j.exmath.2005.11.001

Cited by 15 publications

(3 citation statements)

References 28 publications

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“…There is also the notion of an inner ideal in a Jordan algebra (see [McC71b, Theorem 8] for a description of them for ). The inner ideals are related to the projective homogeneous varieties associated with the group of isometries described in Section 15 and “outer automorphisms” relating these varieties (see [Rac] and [CarrG]).…”

Section: The Ideal Structure Of Freudenthal Algebrasmentioning

confidence: 99%

Albert algebras over and other rings

Garibaldi

Petersson

Racine

2023

Forum of Mathematics, Sigma

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Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative nonassociative algebras and also arise naturally in the context of simple affine group schemes of type $\mathsf {F}_4$ , $\mathsf {E}_6$ , or $\mathsf {E}_7$ . We study these objects over an arbitrary base ring R, with particular attention to the case $R = \mathbb {Z}$ . We prove in this generality results previously in the literature in the special case where R is a field of characteristic different from 2 and 3.

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Section: The Ideal Structure Of Freudenthal Algebrasmentioning

confidence: 99%

Albert algebras over and other rings

Garibaldi

Petersson

Racine

2023

Forum of Mathematics, Sigma

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“…Furthermore, such a variety has a point if and only if this subset consists of circled vertices [BT65, 6.3,1]. In this case, we have an explicit description of the variety associated to vertex 1 in [GC06,7.10]. It is {Kv | v ∈ A with v = 0 and v ♯ = 0}.…”

Section: Real Lie Algebrasmentioning

confidence: 99%

Killing forms of isotropic Lie algebras

Malagon

2012

Journal of Pure and Applied Algebra

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“…We denote by c [ij], where c ∈ C and 1 ≤ i = j ≤ 3, the matrix of the form (4) is a parabolic subgroup of type {α 1 , α 6 } (cf. [13]). The first nontrivial member of the flag is spanned by e 1 , and the second coincides with the summand J 0 (e 3 ) of the Pierce decomposition induced by e 3 (i.e., with the set of elements of J cancelled by e 3 ).…”

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confidence: 99%

Elementary subgroups of isotropic reductive groups

Petrov¹,

Stavrova²

2009

St. Petersburg Math. J.

101

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Abstract. Let G be a not necessarily split reductive group scheme over a commutative ring R with 1. Given a parabolic subgroup P of G, the elementary group E P (R) is defined to be the subgroup of G(R) generated by U P (R) and U P − (R), where U P and U P − are the unipotent radicals of P and its opposite P − , respectively. It is proved that if G contains a Zariski locally split torus of rank 2, then the group E P (R) = E(R) does not depend on P , and, in particular, is normal in G(R). §1. IntroductionLet G be a reductive algebraic group over a commutative ring R with identity. Our aim is to give a definition of an elementary subgroup E(R) of the group of points G(R), generalizing the notion of the elementary subgroup of a split reductive group and other similar concepts, and to show that under some natural restrictions, E(R) is normal in G(R).The notion of the elementary subgroup E n (R) of the general linear group GL n (R) was introduced by Bass [7] (while before it had been used implicitly by Whitehead in the study of homotopy types of CW-complexes) and served as a basis for his construction of algebraic K-theory. In particular, the nonstable K 1 -functor is defined as the quotient GL n (R)/ E n (R), and K 2 as the kernel of a certain central extension of E n (R). The definition of the elementary subgroup involves a fixed basis in R n , but by the Suslin theorem [26], if R is commutative and n ≥ 3, then E n (R) does not depend on the choice of a basis, or, in other words, is normal in GL n (R). Various approaches to this result were discussed, for example, in [25,35].Later on, the elementary subgroup was defined for arbitrary split semisimple groups over R as the subgroup generated by all elementary root unipotents x α (ξ) or, what is the same, by the R-points of the unipotent radical of a Borel subgroup B in G and of the unipotent radical of the opposite Borel subgroup B − (see, e.g., [1,21]). In the same way as in the case of G = GL n , it turns out that when the ranks of all irreducible components of the root system of G are at least 2, the elementary subgroup does not depend on the choice of a Borel subgroup, i.e., is normal in G(R). For the orthogonal and symplectic groups, this fact was proved by Suslin and Kopeȋko [27,18] and by Fu An Li [19], and for arbitrary Chevalley groups by Abe [1] in the case of local rings and by Taddei [29] in the general case (cf. [2]). A simpler proof was given by Hazrat and Vavilov in [15]. The normality of the elementary subgroup in twisted Chevalley groups was proved by Suzuki [28], and by Bak and Vavilov [5]. For classical groups, there are versions of the definition of the elementary subgroup that involve an involution and a "form parameter" (in the sense of Bak). In that case normality was proved by Vaserstein and Hong You [34], and by Bak and Vavilov [6]; see also the paper of the first author on the case of "odd" unitary groups [22]. Certainly, not all classical groups in the sense of Bak can be presented as groups of points of reductive group schemes, but as fo...

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Geometries, the principle of duality, and algebraic groups

Cited by 15 publications

References 28 publications

Albert algebras over and other rings

Albert algebras over and other rings

Killing forms of isotropic Lie algebras

Elementary subgroups of isotropic reductive groups

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