A new construction of Moufang quadrangles of type $E_6, E_7$ and $E_8$
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- by Lien Boelaert and Tom De Medts PDF
- Trans. Amer. Math. Soc. 367 (2015), 3447-3480 Request permission
Abstract:
In the classification of Moufang polygons by J. Tits and R. Weiss, the most intricate case is by far the case of the exceptional Moufang quadrangles of type $E_6$, $E_7$ and $E_8$, and in fact, the construction that they present is ad-hoc and lacking a deeper explanation. We will show how tensor products of two composition algebras can be used to construct these Moufang quadrangles in characteristic different from 2.
As a byproduct, we will obtain a method to construct any Moufang quadrangle in characteristic different from 2 from a module for a Jordan algebra.
References
- B. N. Allison, A class of nonassociative algebras with involution containing the class of Jordan algebras, Math. Ann. 237 (1978), no.Β 2, 133β156. MR 507909, DOI 10.1007/BF01351677
- B. N. Allison, Structurable division algebras and relative rank one simple Lie algebras, Lie algebras and related topics (Windsor, Ont., 1984) CMS Conf. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 1986, pp.Β 139β156. MR 832197
- B. N. Allison, Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras, Trans. Amer. Math. Soc. 306 (1988), no.Β 2, 667β695. MR 933312, DOI 10.1090/S0002-9947-1988-0933312-2
- Bruce Allison, Georgia Benkart, and Yun Gao, Lie algebras graded by the root systems $BC_r,\ r\ge 2$, Mem. Amer. Math. Soc. 158 (2002), no.Β 751, x+158. MR 1902499, DOI 10.1090/memo/0751
- Lien Boelaert and Tom De Medts, Exceptional Moufang quadrangles and structurable algebras, Proc. Lond. Math. Soc. (3) 107 (2013), no.Β 3, 590β626. MR 3100778, DOI 10.1112/plms/pds088
- Tom De Medts, A characterization of quadratic forms of type $E_6,\ E_7$, and $E_8$, J. Algebra 252 (2002), no.Β 2, 394β410. MR 1925144, DOI 10.1016/S0021-8693(02)00064-9
- Tom De Medts, An algebraic structure for Moufang quadrangles, Mem. Amer. Math. Soc. 173 (2005), no.Β 818, vi+99. MR 2109785, DOI 10.1090/memo/0818
- Tom De Medts and Hendrik Van Maldeghem, Moufang generalized polygons, Topics in diagram geometry, Quad. Mat., vol. 12, Dept. Math., Seconda Univ. Napoli, Caserta, 2003, pp.Β 59β126. MR 2066523
- Max-Albert Knus, Alexander Merkurjev, Markus Rost, and Jean-Pierre Tignol, The book of involutions, American Mathematical Society Colloquium Publications, vol. 44, American Mathematical Society, Providence, RI, 1998. With a preface in French by J. Tits. MR 1632779, DOI 10.1090/coll/044
- T. Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005. MR 2104929, DOI 10.1090/gsm/067
- Kevin McCrimmon, A taste of Jordan algebras, Universitext, Springer-Verlag, New York, 2004. MR 2014924
- R. Parimala, J.-P. Tignol, and R. M. Weiss, The Kneser-Tits conjecture for groups with Tits-index $\mathsf {E}_{8,2}^{66}$ over an arbitrary field, Transform. Groups 17 (2012), no.Β 1, 209β231. MR 2891217, DOI 10.1007/s00031-011-9165-2
- W. A. Stein et al., Sage Mathematics Software (Version 4.6.1), The Sage Development Team, 2011, http://www.sagemath.org.
- Tonny A. Springer and Ferdinand D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. MR 1763974, DOI 10.1007/978-3-662-12622-6
- Jacques Tits and Richard M. Weiss, Moufang polygons, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. MR 1938841, DOI 10.1007/978-3-662-04689-0
- Richard M. Weiss, Quadrangular algebras, Mathematical Notes, vol. 46, Princeton University Press, Princeton, NJ, 2006. MR 2177056
Additional Information
- Lien Boelaert
- Affiliation: Department of Mathematics, Ghent University, Krijgslaan 281, S22, B-9000 Gent, Belgium
- MR Author ID: 1035849
- Email: lboelaer@cage.UGent.be
- Tom De Medts
- Affiliation: Department of Mathematics, Ghent University, Krijgslaan 281, S22, B-9000 Gent, Belgium
- MR Author ID: 701084
- ORCID: 0000-0002-9504-5353
- Email: tdemedts@cage.UGent.be
- Received by editor(s): January 25, 2013
- Received by editor(s) in revised form: May 14, 2013
- Published electronically: November 20, 2014
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 3447-3480
- MSC (2010): Primary 17A75, 17A40, 17C40, 20G15, 20G41; Secondary 17C27, 51E12
- DOI: https://doi.org/10.1090/S0002-9947-2014-06195-3
- MathSciNet review: 3314813