Stably Cayley groups in characteristic zero.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1335.20050
author: Mikhail Borovoi
author: Boris Kunyavskiĭ
author: Nicole Lemire
author: Zinovy Reichstein
publication date: 2014-11-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.1329
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 14L35
Mathematics Subject Classification ID: 20G07
Mathematics Subject Classification ID: 14E05
zbMATH DE Number: 6369076
zbMATH Keywords: linear algebraic groups
zbMATH Keywords: stably Cayley groups
zbMATH Keywords: equivariant birational isomorphisms
zbMATH Keywords: character lattices
MaRDI profile type: Publication
OpenAlex ID: W3100009961
Wikidata QID: Q117370911
arXiv ID: 1207.1329
DOI: 10.1093/IMRN/RNT123
published in: IMRN. International Mathematics Research Notices
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title: Stably Cayley groups in characteristic zero.
review text: Let \(G\) be a connected affine algebraic group over a field \(k\) and let \(\mathfrak g\) be the Lie algebra of \(G\). Consider the action of \(G\) on itself by conjugation and on \(\mathfrak g\) via the adjoint representation. If there is a \(G\)-equivariant birational isomorphism \(G\dashrightarrow\mathfrak g\) defined over \(k\), then \(G\) is called a Cayley group over \(k\). If \(G\times\mathbb G_m^r\) is Cayley over \(k\) for some \(r\geqslant 0\), then \(G\) is called stably Cayley over \(k\). These notions were introduced by \textit{N. Lemire, V. L. Popov} and \textit{Z. Reichstein} [in J. Am. Math. Soc. 19, No. 4, 921-967 (2006; Zbl 1103.14026)], where, in particular, Cayley and stably Cayley simple groups have been classified in the case where \(k\) is an algebraically closed field of characteristic zero.NEWLINENEWLINE In the paper under review, the authors study reductive Cayley groups over an arbitrary field \(k\). They give a criterion for a reductive group \(G\) to be stably Cayley, formulated in terms of its character lattice, and obtain the following classifications:NEWLINENEWLINE Theorem 1. Let \(k\) be a field of characteristic \(0\) and let \(G\) be an absolutely simple \(k\)-group. Then the following conditions are equivalent: (a) \(G\) is stably Cayley over \(k\); (b) \(G\) is an arbitrary \(k\)-form of one of the following groups: \(\text{SL}_3\), \(\text{PGL}_n\) (\(n=2\) or \(n\geqslant 3\) odd), \(\text{SO}_n\) (\(n\geqslant 5\)), \(\text{Sp}_{2n}\) (\(n\geqslant 1\)), \(\text G_2\), or an inner \(k\)-form of \(\text{PGL}_n\) (\(n\geqslant 4\) even).NEWLINENEWLINE Theorem 2. Let \(k\) be a field of characteristic \(0\) and let \(G\) be a simple (but not necessarily absolutely simple) \(k\)-group. Then the following conditions are equivalent: (a) \(G\) is stably Cayley over \(k\); (b) \(G\) is isomorphic to \(R_{l/k}(G_1)\), where \(l/k\) is a finite field extension and \(G_1\) is either a stably Cayley absolutely simple group over \(l\) (i.…
reviewed by: Vladimir L. Popov
description: scientific article; zbMATH DE number 6369076