Commuting $U$-operators in Jordan algebras
HTML articles powered by AMS MathViewer
- by José A. Anquela, Teresa Cortés and Holger P. Petersson PDF
- Trans. Amer. Math. Soc. 366 (2014), 5877-5902 Request permission
Abstract:
For elements $x,y$ in a non-degenerate non-unital Jordan algebra over a commutative ring, the relation $x \circ y = 0$ is shown to imply that the $U$-operators of $x$ and $y$ commute: $U_xU_y = U_yU_x$. The proof rests on the Zel$’$manov-McCrimmon classification of strongly prime quadratic Jordan algebras.References
- Tom De Medts and Yoav Segev, A course on Moufang sets, Innov. Incidence Geom. 9 (2009), 79–122. MR 2658895, DOI 10.2140/iig.2009.9.79
- Tom De Medts and Richard M. Weiss, Moufang sets and Jordan division algebras, Math. Ann. 335 (2006), no. 2, 415–433. MR 2221120, DOI 10.1007/s00208-006-0761-8
- John R. Faulkner, Octonion planes defined by quadratic Jordan algebras, Memoirs of the American Mathematical Society, No. 104, American Mathematical Society, Providence, R.I., 1970. MR 0271180
- John R. Faulkner, Finding octonion algebras in associative algebras, Proc. Amer. Math. Soc. 104 (1988), no. 4, 1027–1030. MR 931729, DOI 10.1090/S0002-9939-1988-0931729-9
- Nathan Jacobson, Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, Vol. XXXIX, American Mathematical Society, Providence, R.I., 1968. MR 0251099
- N. Jacobson, Lectures on quadratic Jordan algebras, Tata Institute of Fundamental Research Lectures on Mathematics, No. 45, Tata Institute of Fundamental Research, Bombay, 1969. MR 0325715
- Nathan Jacobson, Structure theory of Jordan algebras, University of Arkansas Lecture Notes in Mathematics, vol. 5, University of Arkansas, Fayetteville, Ark., 1981. MR 634508
- Serge Lang, Algebra, 3rd ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. MR 1878556, DOI 10.1007/978-1-4613-0041-0
- Robert E. Lewand and Kevin McCrimmon, Macdonald’s theorem for quadratic Jordan algebras, Pacific J. Math. 35 (1970), 681–706. MR 299648
- Kevin McCrimmon, The Freudenthal-Springer-Tits constructions of exceptional Jordan algebras, Trans. Amer. Math. Soc. 139 (1969), 495–510. MR 238916, DOI 10.1090/S0002-9947-1969-0238916-9
- Kevin McCrimmon, The Freudenthal-Springer-Tits constructions revisited, Trans. Amer. Math. Soc. 148 (1970), 293–314. MR 271181, DOI 10.1090/S0002-9947-1970-0271181-3
- Kevin McCrimmon, Quadratic Jordan algebras and cubing operations, Trans. Amer. Math. Soc. 153 (1971), 265–278. MR 268239, DOI 10.1090/S0002-9947-1971-0268239-2
- Kevin McCrimmon, Nonassociative algebras with scalar involution, Pacific J. Math. 116 (1985), no. 1, 85–109. MR 769825
- Kevin McCrimmon, A taste of Jordan algebras, Universitext, Springer-Verlag, New York, 2004. MR 2014924
- Kevin McCrimmon and Ephim Zel′manov, The structure of strongly prime quadratic Jordan algebras, Adv. in Math. 69 (1988), no. 2, 133–222. MR 946263, DOI 10.1016/0001-8708(88)90001-1
- Holger P. Petersson, An embedding theorem for reduced Albert algebras over arbitrary fields, to appear in Communications in Algebra.
- Holger P. Petersson, On linear and quadratic Jordan division algebras, Math. Z. 177 (1981), no. 4, 541–548. MR 624231, DOI 10.1007/BF01219086
- Holger P. Petersson and Michel L. Racine, Jordan algebras of degree $3$ and the Tits process, J. Algebra 98 (1986), no. 1, 211–243. MR 825144, DOI 10.1016/0021-8693(86)90024-4
- 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
- Armin Thedy, $z$-closed ideals of quadratic Jordan algebras, Comm. Algebra 13 (1985), no. 12, 2537–2565. MR 811523, DOI 10.1080/00927878508823290
- 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
Additional Information
- José A. Anquela
- Affiliation: Departamento de Matemáticas, Universidad de Oviedo, E-33007 Oviedo, Spain
- Email: anque@orion.ciencias.uniovi.es
- Teresa Cortés
- Affiliation: Departamento de Matemáticas, Universidad de Oviedo, E-33007 Oviedo, Spain
- Email: cortes@orion.ciencias.uniovi.es
- Holger P. Petersson
- Affiliation: Fakultät für Mathematik und Informatik, FernUniversität in Hagen, D-58084 Hagen, Germany
- MR Author ID: 138575
- Email: holger.petersson@fernuni-hagen.de
- Received by editor(s): August 20, 2012
- Received by editor(s) in revised form: December 13, 2012, and December 17, 2012
- Published electronically: July 17, 2014
- Additional Notes: The research of the first two authors was partially supported by the Spanish Ministerio de Economía y Competitividad and Fondos FEDER, MTM2010-16153, and MTM2013-40841-P
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 366 (2014), 5877-5902
- MSC (2010): Primary 17C10; Secondary 20B22, 20E42, 17C40, 17C60
- DOI: https://doi.org/10.1090/S0002-9947-2014-06054-6
- MathSciNet review: 3256187