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A new construction of Moufang quadrangles of type $ E_6, E_7$ and $ E_8$


Authors: Lien Boelaert and Tom De Medts
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
Published electronically: November 20, 2014
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [A1] B. N. Allison, A class of nonassociative algebras with involution containing the class of Jordan algebras, Math. Ann. 237 (1978), no. 2, 133-156. MR 507909 (81h:17003), https://doi.org/10.1007/BF01351677
  • [A2] 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. 139-156. MR 832197 (87j:17001)
  • [A3] B. N. Allison, Tensor products of composition algebras, Albert forms and some exceptional simple Lie algebras, Trans. Amer. Math. Soc. 306 (1988), no. 2, 667-695. MR 933312 (89e:17004), https://doi.org/10.2307/2000817
  • [ABG] B. N. Allison, G. Benkart, and Y. Gao, Lie algebras graded by the root systems $ BC_r,\ r\ge 2$, Mem. Amer. Math. Soc. 158 (2002), no. 751, x+158. MR 1902499 (2003h:17038), https://doi.org/10.1090/memo/0751
  • [BD] L. Boelaert and T. De Medts, Exceptional Moufang quadrangles and structurable algebras, Proc. London Math. Soc. (3) 107 (2013), no. 3, 590-626. MR 3100778
  • [D1] T. De Medts, A characterization of quadratic forms of type $ E_6,\ E_7$, and $ E_8$, J. Algebra 252 (2002), no. 2, 394-410. MR 1925144 (2003f:11046), https://doi.org/10.1016/S0021-8693(02)00064-9
  • [D2] T. De Medts, An algebraic structure for Moufang quadrangles, Mem. Amer. Math. Soc. 173 (2005), no. 818, vi+99. MR 2109785 (2005k:51003), https://doi.org/10.1090/memo/0818
  • [DV] T. De Medts and H. Van Maldeghem, Moufang generalized polygons, Topics in diagram geometry, Quad. Mat., vol. 12, Dept. Math., Seconda Univ. Napoli, Caserta, 2003, pp. 59-126. MR 2066523 (2005d:51007)
  • [KMRT] M.-A. Knus, A. Merkurjev, M. Rost, and J.-P. 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 (2000a:16031)
  • [L] T. Y. Lam, Introduction to quadratic forms over fields, Graduate Studies in Mathematics, vol. 67, American Mathematical Society, Providence, RI, 2005. MR 2104929 (2005h:11075)
  • [M] K. McCrimmon, A taste of Jordan algebras, Universitext, Springer-Verlag, New York, 2004. MR 2014924 (2004i:17001)
  • [PTW] 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. 1, 209-231. MR 2891217, https://doi.org/10.1007/s00031-011-9165-2
  • [Sage] W.A. Stein et al., Sage Mathematics Software (Version 4.6.1), The Sage Development Team, 2011, http://www.sagemath.org.
  • [SV] T. A. Springer and F. D. Veldkamp, Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2000. MR 1763974 (2001f:17006)
  • [TW] J. Tits and R. M. Weiss, Moufang polygons, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. MR 1938841 (2003m:51008)
  • [W] R. M. Weiss, Quadrangular algebras, Mathematical Notes, vol. 46, Princeton University Press, Princeton, NJ, 2006. MR 2177056 (2007a:17010)

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Additional Information

Lien Boelaert
Affiliation: Department of Mathematics, Ghent University, Krijgslaan 281, S22, B-9000 Gent, Belgium
Email: lboelaer@cage.UGent.be

Tom De Medts
Affiliation: Department of Mathematics, Ghent University, Krijgslaan 281, S22, B-9000 Gent, Belgium
Email: tdemedts@cage.UGent.be

DOI: https://doi.org/10.1090/S0002-9947-2014-06195-3
Keywords: Moufang polygons, Moufang quadrangles, composition algebras, octonion algebras, quadrangular algebras, Jordan algebras, structurable algebras, $J$-ternary algebras, linear algebraic groups, exceptional groups, $E_6$, $E_7$, $E_8$
Received by editor(s): January 25, 2013
Received by editor(s) in revised form: May 14, 2013
Published electronically: November 20, 2014
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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