Cayley groups

Lemire, Nicole; Popov, Vladimir L.; Reichstein, Zinovy

doi:10.1090/s0894-0347-06-00522-4

Cited by 39 publications

(57 citation statements)

References 30 publications

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“…Next we turn our attention to classifying stably Cayley simple groups over an arbitrary field k of characteristic zero. The following results extend [20,Theorem 1.28], where k is assumed to be algebraically closed.…”

Section: Introductionsupporting

confidence: 64%

“…These notions were introduced by Lemire, Popov and Reichstein [20]; for a more detailed discussion and numerous classical examples, we refer the reader to [20,Introduction]. The main results of [20] are the classifications of Cayley and stably Cayley simple groups in the case where the base field k is algebraically closed and of characteristic 0. The goal of this paper is to extend some of these results to the case where k is an arbitrary field of characteristic 0.…”

Section: Introductionmentioning

confidence: 99%

“…Example 1.1. If k is algebraically closed and G is a reductive k-group, then by [20,Theorem 1.27] G is stably Cayley if and only if its character lattice is quasi-permutation; see Definition 2.1. Example 1.2.…”

Section: Introductionmentioning

confidence: 99%

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Stably Cayley Groups in Characteristic Zero

Borovoi

Kunyavskiı̆

Lemire

et al. 2013

International Mathematics Research Notices

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A linear algebraic group G over a field k is called a Cayley group if it admits a Cayley map, i.e., a G-equivariant birational isomorphism over k between the group variety G and its Lie algebra. A Cayley map can be thought of as a partial algebraic analogue of the exponential map. A prototypical example is the classical "Cayley transform" for the special orthogonal group SOn defined by Arthur Cayley in 1846. A linear algebraic group G is called stably Cayley if G × G r m is Cayley for some r ≥ 0. Here G r m denotes the split r-dimensional k-torus. These notions were introduced in 2006 by Lemire, Popov and Reichstein, who classified Cayley and stably Cayley simple groups over an algebraically closed field of characteristic zero.In this paper we study reductive Cayley groups over an arbitrary field k of characteristic zero. Our main results are a criterion for a reductive group G to be stably Cayley, formulated in terms of its character lattice, and a classification of stably Cayley simple groups.2010 Mathematics Subject Classification. Primary 20G15, 20C10.

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Section: Introductionsupporting

confidence: 64%

Section: Introductionmentioning

confidence: 99%

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Stably Cayley Groups in Characteristic Zero

Borovoi

Kunyavskiı̆

Lemire

et al. 2013

International Mathematics Research Notices

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“…Let N be the normalizer of t in G. We endow t⊕t with the natural N -module structure. Let G × N (t ⊕ t) be the algebraic homogeneous vector G-bundle over G/N with fiber t ⊕ t, see [Se,§2], [PV 2 , 4.8], [LPR,2.17]. Denote by g * t the image of point (g, t) ∈ G × (t ⊕ t) under the natural projection G × (t ⊕ t) → G × N (t ⊕ t).…”

Section: 2mentioning

confidence: 99%

Irregular and Singular Loci of Commuting Varieties

Popov

2008

Transformation Groups

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Let C be the commuting variety of the Lie algebra g of a connected noncommutative reductive algebraic group G over an algebraically closed field of characteristic zero. Let C sing be the singular locus of C and let C irr be the locus of points whose Gstabilizers have dimension > rk G. We prove that: (a) C sing is a nonempty subset of C irr ; (b) codim C C irr = 5 − max l(a) where the maximum is taken over all simple ideals a of g and l(a) is the "lacety" of a; and (c) if t is a Cartan subalgebra of g and α ∈ t * a root of g with respect to t, then G(Ker α × Ker α) is an irreducible component of C irr of codimension 4 in C. This yields the bound codim C C sing 5−max l(a) and, in particular, codim C C sing 2. The latter may be regarded as an evidence in favor of the known longstanding conjecture that C is always normal. We also prove that the algebraic variety C is rational.

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“…Our goal in this section is to prove the following variant of [LPR,Lemma 2.17], which will be repeatedly used in the sequel. Here, as usual, RMaps H (Z, R) denotes the k(Z) H -algebra of H -equivariant rational maps Z R.…”

Section: Homogeneous Fiber Spacesmentioning

confidence: 99%