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St. Petersburg Mathematical Journal

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Identity with constants in a Chevalley group of type $ {\mathrm F}_4$

Authors: V. Nesterov and A. Stepanov
Translated by: the authors
Original publication: Algebra i Analiz, tom 21 (2009), nomer 5.
Journal: St. Petersburg Math. J. 21 (2010), 819-823
MSC (2010): Primary 20G07
Published electronically: July 15, 2010
MathSciNet review: 2604568
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Abstract | References | Similar Articles | Additional Information

Abstract: N. L. Gordeev proved that a generalized group identity holds in Chevalley groups with multiply laced root systems. It was also shown that a stronger identity is valid for the Chevalley groups of types $ \mathrm{B}_l$ and $ \mathrm{C}_l$. In the present paper, it is proved that this strong identity is fulfilled in Chevalley groups of type $ \mathrm{F}_4$ and fails to be true in Chevalley groups of type $ \mathrm{G}_2$. The main result of the paper is the last ingredient in the proof of the claim that the lattice of intermediate subgroups between $ G(\mathrm{F}_4,R)$ and $ G(\mathrm{F}_4,A)$ is standard for an arbitrary pair of rings $ R\subseteq A$ with 2 invertible.

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  • 1. I. Z. Golubchik and A. V. Mikhalëv, Generalized group identities in classical groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 114 (1982), 96–119, 219 (Russian). Modules and algebraic groups. MR 669562
  • 2. N. A. Vavilov and A. V. Stepanov, Overgroups of semisimple groups, Vestn. Samar. Gos. Univ. Estestvennonauchn. Ser. 3 (2008), 51–95 (Russian, with English and Russian summaries). MR 2473730
  • 3. N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968 (French). MR 0240238
  • 4. Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, Conn., 1968. Notes prepared by John Faulkner and Robert Wilson. MR 0466335
  • 5. G. M. Tomanov, Generalized group identities in linear groups, Mat. Sb. (N.S.) 123(165) (1984), no. 1, 35–49 (Russian). MR 728928
  • 6. N. A. Vavilov, The geometry of long root subgroups in Chevalley groups, Vestnik Leningrad. Univ. Mat. Mekh. Astronom. vyp. 1 (1988), 8–11, 116 (Russian, with English summary); English transl., Vestnik Leningrad Univ. Math. 21 (1988), no. 1, 5–10. MR 946454
  • 7. N. L. Gordeev, Freedom in conjugacy classes of simple algebraic groups and identities with constants, Algebra i Analiz 9 (1997), no. 4, 63–78; English transl., St. Petersburg Math. J. 9 (1998), no. 4, 709–723. MR 1604024
  • 8. A. V. Stepanov, Free product subgroups between Chevalley groups $ {\mathrm G}(\Phi,F)$ and $ {\mathrm G}(\Phi,F[t])$, Preprint:, 2007, to be published in J. Algebra.
  • 9. -, Subring subgroups in Chevalley groups with doubly laced root systems, Preprint:, 2009, to be published in J. Algebra.

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Additional Information

V. Nesterov
Affiliation: Baltic State Technical University, 1-st Krasnoarmeiskaya Street 1, St. Petersburg 190005, Russia

A. Stepanov
Affiliation: St. Petersburg Electrotechnical University, Professor Popov Street 5, St. Petersburg 197376, Russia

Keywords: Group identity, Chevalley group, multiply laced root system
Received by editor(s): September 8, 2008
Published electronically: July 15, 2010
Additional Notes: The second author was supported by RFBR (grant no. 08-01-00756-a).
Article copyright: © Copyright 2010 American Mathematical Society