Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-26T17:40:39.298Z Has data issue: false hasContentIssue false
  • English
  • Français

Order 3 Elements in G2 and Idempotents in Symmetric Composition Algebras

Published online by Cambridge University Press:  20 November 2018

Alberto Elduque
Alberto Elduque*
Affiliation:
Departamento de Matemácas, Universidad de Zaragoza, 50009 Zaragoza, Spain, e-mail: elduque@unizar.es
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Order three elements in the exceptional groups of type ${{G}_{2}}$ are classified up to conjugation over arbitrary fields. Their centralizers are computed, and the associated classification of idempotents in symmetric composition algebras is obtained. Idempotents have played a key role in the study and classification of these algebras.

Over an algebraically closed field, there are two conjugacy classes of order three elements in ${{G}_{2}}$ in characteristic not 3 and four of them in characteristic 3. The centralizers in characteristic 3 fail to be smooth for one of these classes.

Keywords

17A7514L1517B2520G15symmetric composition algebraOkubo algebraautomorphism groupcentralizeridempotent
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 5 , 01 October 2018 , pp. 1038 - 1075
Copyright
Copyright © Canadian Mathematical Society 2018

References

[AEMN02] Alberca-Bjerregaard, P., Elduque, A., Martin-Gonzalez, C., and Navarro-Mârquez, F. J., On the Cartan-Jacobson theorem. J. Algebra 250 (2002), no. 2, 397–407. http://dx.doi.Org/10.1006/jabr.2001.9084Google Scholar
[AE16] Castillo-Ramirez, A. and Elduque, A., Some special features of Cayley algebras, and G2, in low characteristics. J. Pure Appl. Algebra 220 (2016) no. 3, 1188–1205. http://dx.doi.Org/10.1016/j.jpaa.2015.08.015Google Scholar
[CEKT13] Chernousov, V., Elduque, A., Knus, M.-A., and Tignol, J.-P., Algebraic groups of type D4, triality, and composition algebras. Doc. Math. 18 (2013), 413–468.Google Scholar
[DM09] Draper, C. and Martin-Gonzalez, C., Gradings on the Albert algebra and on f4. Rev. Mat. Iberoam. 25 (2009), no. 3, 841–908.Google Scholar
[Eld97] Elduque, A., Symmetric composition algebras. J. Algebra 196 (1997), no. 1, 282–300. http://dx.doi.Org/10.1006/jabr.1997.7071Google Scholar
[Eld99] Elduque, A., Okubo algebras in characteristic 3 and their automorphisms. Comm. Algebra 27 (1999), no. 6, 3009–3030. http://dx.doi.org/10.1080/00927879908826607Google Scholar
[Eld09] Elduque, A., Gradings on symmetric composition algebras. J. Algebra 322 (2009), no. 10, 3542–3579. http://dx.doi.Org/10.1016/j.jalgebra.2009.07.031Google Scholar
[Eldl5] Elduque, A., Okubo algebras: automorphisms, derivations and idempotents. Lie algebras and related topics, Contemp. Math., 652, American Mathematical Society, Providence, RI, 2015, pp. 61-73. http://dx.doi.org/10.1090/conm/652/12953Google Scholar
[EK13] Elduque, A. and Kochetov, M., Gradings on simple Lie algebras. Mathematical Surveys and Monographs, 189, American Mathematical Society, Providence, RI, 2013. http://dx.doi.Org/10.1090/surv/189Google Scholar
[EM91] Elduque, A. and Myung, H.-C., Flexible composition algebras and Okubo algebras. Comm. Algebra 19 (1991), no. 4, 1197–1227. http://dx.doi.org/10.1080/00927879108824198Google Scholar
[EM93] Elduque, A. and Myung, H.-C., On flexible composition algebras. Comm. Algebra 21 (1993), no. 7, 2481–2505. http://dx.doi.org/10.1080/00927879308824688Google Scholar
[EM95] Elduque, A. and Myung, H.-C., Colour algebras and Cayley-Dickson algebras. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), no. 6, 1287–1303. http://dx.doi.org/!0.1017/S0308210500030511Google Scholar
[EP96] Elduque, A., Pérez-Izquierdo, J.-M., Composition algebras with associative bilinear form. Comm. Algebra 24 (1996), no. 3, 1091–1116. http://dx.doi.org/10.1080/00927879608825625Google Scholar
[Jac58] Jacobson, N., Composition algebras and their automorphisms. Rend. Circ. Mat. Palermo (2) 7 (1958), 55–80. http://dx.doi.org/10.1007/BF02854388Google Scholar
[Kac90] Kac, V. G., Infinite-dimensional Lie algebras. Third éd., Cambridge University Press, Cambridge, 1990. http://dx.doi.org/10.1017/CBO9780511626234Google Scholar
[KMRT98] Knus, M.-A., Merkurjev, A., Rost, M., and Tignol, J.-P., The book of involutions. American Mathematical Society Colloquium Publications, 44, American Mathematical Society, Providence, RI, 1998. http://dx.doi.Org/10.1090/colI/044Google Scholar
[Oku78] Okubo, S., Pseudo-quaternion and pseudo-octonion algebras. Hadronic J. 1 (1978), no. 4, 1250–1278.Google Scholar
[OO81a] Okubo, S. and Osborn, J. M. Algebras with nondegenerate associative symmetric bilinear forms permitting composition. Comm. Algebra 9 (1981) no. 12,1233-1261. http://dx.doi.Org/10.1080/00927878108822644Google Scholar
[OO81b] Algebras with nondegenerate associative symmetric bilinear forms permitting composition. II. Comm. Algebra 9 (1981), no. 20, 2015–2073. http://dx.doi.Org/10.1080/00927878108822695Google Scholar
[PL15] Pepin Lehalleur, S., Subgroups of maximal rank of reductive groups. In: Autour des schémas en groupes. Vol. III. Panoramas et Synthèses, 47, Société Mathématique de France, Paris, 2015, pp. 147-172.Google Scholar
[Pet69] Petersson, H. P., Eine Identitàt fùnften Grades, der gewisse Isotope von Kompositions-Algebren genûgen. Math. Z. 109(1969) 217–238. http://dx.doi.org/10.1007/BF01111407Google Scholar
[Pla68] Platonov, V. P., The theory of algebraic linear groups and periodic groups. American Society Translations (2) 69 (1968), 61–110.Google Scholar
[Sch95] Schafer, R. D., An introduction to nonassociative algebras, corrected reprint of the 1966 original. Dover Publications, Inc., New York, 1995.Google Scholar
[SerO6] Serre, J. P., Coordonnées de Kac. Oberwolfach Reports 3 (2006), 1787–1790.Google Scholar
[SprO9] Springer, T. A., Linear algebraic groups. Second éd., Modern Birkhâuser Classics. Birkhâuser Boston, Inc., Boston, MA, 2009.Google Scholar
[SVOO] Springer, T. A. and Veldkamp, E. D., Octonions, Jordan algebras and exceptional groups. Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. http://dx.doi.org/10.1007/978-3-662-12622-6Google Scholar
[Ste68] Steinberg, R., Lectures on Chevalley groups. Yale University, New Haven, Conn., 1968.Google Scholar
[Tit87] Tits, J., Unipotent elements and parabolic subgroups of reductive groups. II. In: Algebraic groups Utrecht 1986, Lecture Notes in Math., 1271, Springer, Berlin, 1987, pp. 265-284. http://dx.doi.org/10.1007/BFb0079243Google Scholar
[Wat79] Waterhouse, W. C., Introduction to affine group schemes. Graduate Texts in Mathematics, 66, Springer-Verlag, New York-Berlin, 1979.Google Scholar
[ZSSS82] Zhevlakov, K. A., Slin'ko, A. M., Shestakov, I. P., and Shirshov, A. I., Rings that are nearly associative. Pure and Applied Mathematics, 104, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York-London, 1982.Google Scholar