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The Role of Nonassociative Algebra in Projective Geometry

About this Title

John R. Faulkner, University of Virginia, Charlottesville, VA

Publication: Graduate Studies in Mathematics
Publication Year: 2014; Volume 159
ISBNs: 978-1-4704-1849-6 (print); 978-1-4704-1933-2 (online)
DOI: https://doi.org/10.1090/gsm/159
MathSciNet review: MR3237696
MSC: Primary 51-02; Secondary 17A75, 17D05, 51Axx, 51C05, 51E24

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Front/Back Matter

Chapters

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References
  • P. Abramenko and K. S. Brown, Buildings: Theory and Applications, Grad. Texts in Math. 248, Springer-Verlag, New York, 2008.
  • B. N. Allison, A class of nonassociative algebras with involution containing the class of Jordan algebras, Math. Ann. 237 (1978), 133–156.
  • B. N. Allison and J. R. Faulkner, Nonassociative coefficient algebras for Steinberg unitary Lie algebras, Jour. Algebra 161 (1993), 1–19.
  • E. Artin, Geometric Algebra, Interscience Publishers, Inc., New York, 1957.
  • G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Annals of Math., 2nd series, 37 (1936), 823–843.
  • L. Boelaert, From the Moufang World to the Structurable World and Back Again, Dissertation Universiteit Gent, 2013.
  • R. H. Bruck and E. Kleinfeld, The structure of alternative rings, Proc. Amer. Math. Soc. 2 (1951), 878–890.
  • F. Buekenhout, editor, Handbook of Incidence Geometry: Buildings and Foundations, Elsevier, Amsterdam, New York, 1995.
  • J. Dieudonné, La Géométrie des Groupes Classiques, 2nd. ed., Springer-Verlag, Berlin, Heidelberg, New York, 1963.
  • C. Chevalley and R. D. Schafer, The exceptional simple Lie algebras $F_{4}$ and $E_{6}$, Proc. Nat. Akad. Sci. U.S.A. 36 (1950), 137–141.
  • H. S. M. Coxeter, The Real Projective Plane, 2nd. ed., Cambridge University Press, Cambridge, 1955.
  • J. R. Faulkner, Octonion planes defined by quadratic Jordan algebras, Mem. Amer. Math. Soc. 104 (1970), 1–71.
  • J. R Faulkner, On the geometry of inner ideals, Jour. Algebra 26 (1973), 1–9.
  • J. R. Faulkner, Groups with Steinberg relations and coordinatization of polygonal geometries, Mem. Amer. Math. Soc. 185 (1977), 1–134.
  • J. R. Faulkner, Barbilian planes, Geom. Dedicata 30 (1989), 125–181.
  • J. R. Faulkner, Structurable triples, Lie triples, and symmetric spaces, Forum Math. 6 (1994), 637–650.
  • J. R. Faulkner, Projective remoteness planes, Geom. Dedicata 60 (1996), 237–275.
  • H. Freudenthal, Beziehungen der $E_{7}$ und $E_{8}$ zur Oktavenebene. VIII, Nederl. Akad. Wetensch. Proc. Ser. A 62 $=$ Indag. Math. 21 (1959), 447–465.
  • H. Freudenthal, Symplektische und metasymplektische Geometrien, Algebraical and Topological Foundations of Geometry (Proc. Colloq., Utrecht, 1959), Pergamon, Oxford, 1962, 29–33.
  • D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea Pub. Co., New York, 1952.
  • J. Humphreys, Introduction to Lie algebras and Representation Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1972.
  • N. Jacobson, Structure and Representations of Jordan Algebras, American Math. Soc. Colloq. Pub. 39, Providence, 1968.
  • N. Jacobson, Basic Algebra II, W. H. Freeman and Company, San Francisco, 1980.
  • E. Kleinfeld, Simple alternative rings, Ann. of Math. 58 (1953), 544–547.
  • O. Loos, Symmetric Spaces, Vol. I, W. A. Benjamin, New York, 1969.
  • K. McCrimmon, The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras, Trans. Amer. Math. Soc. 139 (1969), 495–510.
  • K. McCrimmon, A Taste of Jordan Algebras, Springer-Verlag, 2004.
  • R. Moufang, Alternativekörper und der Satz vom vollständigen Vierseit, Abh. Math. Sem. Hamburg 9 (1933), 207–234.
  • G. Pickert, Projektive Ebenen, Springer-Verlag, Berlin, Heidelberg, New York, 1955.
  • B. A. Rosenfeld, Geometrical Interpretation of the Compact Simple Lie Groups of the Class E (Russian), Dokl. Akad. Nauk. SSSR 106 (1956), 600–603.
  • B. A. Rosenfeld, Geometry of Lie Groups, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
  • H. Salzmann, D. Betten, T. Grundhoefer, H. Hähl, R. Löwen, and M. Stroppel, Compact Projective Planes: With an Introduction to Octonion Geometry, De Gruyter, Berlin, 1995.
  • R. Schafer, An Introduction to Nonassociative Algebras, Academic Press, New York, London, 1966.
  • L. A. Skornyakov, Alternative fields (Russian), Ukrain. Mat. Žurnal 2 (1950), 70–85.
  • T. A. Springer, The projective octave plane, Nederl. Akad. Wetensch. Proc. Ser. A, 63 $=$ Indag. Math. 22 (1960), 74–101.
  • T. A. Springer, On the geometric algebra of the octave plane, Nederl. Akad. Wetensch. Proc. Ser. A, 65 $=$ Indag. Math. 24 (1962), 451–468.
  • T. A. Springer, Jordan Algebras and Algebraic Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 75, Springer-Verlag, Berlin, Heidelberg, New York, 1973.
  • J. Tits, Étude géométrique d’une classe d’espaces homogènes, C. R. Acad. Sci. Paris 239 (1954), 466–468.
  • J. Tits, Groupes semi-simple complexes et géométrie projective, Séminaire N. Bourbaki 1954–1956, exp. 112, 115–125.
  • J. Tits, Sur certaines classes d’espaces homogènes de groupes de Lie, Mém. Acad. Roy. Belg., 29, 268 pp.
  • J. Tits, Sur la géométrie des $R$-espaces, Journ. de Math., Série 9, 36 (1957), 17–38.
  • J. Tits, Algèbres alternatives, algèbres de Jordan et algèbres de Lie exceptionnelles, Nederl. Akad. Wetensch. Proc. Ser. A 69 $=$ Indag. Math. 28 (1966), 223–237.
  • J. Tits, Buildings of Spherical Type and Finite BN-pairs, Lecture Notes in Mathematics, 386, Springer-Verlag, New York, Heidelberg, Berlin, 1974.
  • J. Tits, Non-existence de certains, polygones généralisés, I, Inventiones Math. 36 (1976), 275–284; II, Inventiones Math. 51 (1979), 267–269.
  • J. Tits, Endliche Spiegelungsgruppen, die als Weylgruppen auftreten, Inventiones Math. 43 (1977), 283–295.
  • J. Tits and R. Weiss, Moufang Polygons, Springer-Verlag, Berlin, Heidelberg, New York, 2002.
  • O. Veblen and J. Young, Projective Geometry, Vol. I, Ginn, Boston, 1910.
  • F. Veldkamp, Projective planes over rings of stable rank 2, Geom. Dedicata 11 (1981), 285–308.
  • E. B. Vinberg, Construction of the exceptional simple Lie algebras, Trudy Sem. Vekt. Tenz. Anal. 13 (1966), 7–9; Amer. Math. Soc. Transl. Ser. 2, 213, Lie groups and invariant theory, 241–242, Amer. Math. Soc., Providence, RI, 2005.
  • R. Weiss, The nonexistence of certain Moufang polygons, Inventiones Math. 51 (1979), 261–266.
  • C. Weibel, Survey of non-Desarguesian planes, Notices Amer. Math. Soc. 54 (2007), 1294–1303.
  • K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov, and A. I. Shirshov, translated by H. F. Smith, Rings That Are Nearly Associative, Academic Press, New York, 1982.