Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Jordan bimodules over the superalgebras $ P(n)$ and $ Q(n)$


Authors: Consuelo Martínez, Ivan Shestakov and Efim Zelmanov
Journal: Trans. Amer. Math. Soc. 362 (2010), 2037-2051
MSC (2000): Primary 17C70; Secondary 17C55, 17B10, 17B60
DOI: https://doi.org/10.1090/S0002-9947-09-04997-6
Published electronically: November 13, 2009
MathSciNet review: 2574886
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We extend the Jacobson's Coordinatization theorem to Jordan superalgebras. Using it we classify Jordan bimodules over superalgebras of types $ Q(n)$ and $ JP(n)$, $ n \geq 3$. Then we use the Tits-Kantor-Koecher construction and representation theory of Lie superalgebras to treat the remaining case $ Q(2)$.


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

  • 1. N. Jacobson, Structure and Representation of Jordan algebras, Amer. Math. Soc. Providence, R.I., 1969. MR 0251099 (40:4330)
  • 2. V.G. Kac, Lie Superalgebras, Advances in Math. 26 no.1 (1977), 8-96. MR 0486011 (58:5803)
  • 3. V.G. Kac, Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras, Comm. in Algebra 5 (13) (1977), 1375-1400. MR 0498755 (58:16806)
  • 4. I.L. Kantor, Jordan and Lie superalgebras defined by Poisson brackets, Algebra and Analysis, 55-79, 1989. Amer. Math. Soc. Transl. Ser. (2)151, 1992. MR 1191172 (93j:17004)
  • 5. I. Kaplansky, Superalgebras, Pacific J. of Math. 86 (1980), 93-98. MR 586871 (81j:17006)
  • 6. I. Kaplansky, Graded Jordan Algebras I, Preprint.
  • 7. O. Loos, Jordan pairs, Lecture Notes in Mathematics, vol. 460, Springer-Verlag, Berlin and New York, 1975. MR 0444721 (56:3071)
  • 8. M.C. López-Diaz and I. P. Shestakov, Representations of exceptional simple Jordan superalgebras of characteristic $ 3$, Comm. in Algebra 33, (2005), 331-337. MR 2128489 (2006c:17042)
  • 9. C. Martínez and E. Zelmanov, Simple finite-dimensional Jordan Superalgebras in prime characteristic, Journal of Algebra 236 no.2 (2001), 575-629. MR 1813492 (2002e:17042)
  • 10. C. Martınez and E. Zelmanov, Specializations of Simple Jordan Superalgebras, Canad. Math. Bull. 45 (4) (2002), 653-671. MR 1941232 (2003k:17040)
  • 11. C. Martınez and E. Zelmanov, Unital bimodules over the simple Jordan superalgebra $ D(t)$, Trans. Amer. Math. Soc. 358 no.8 (2006), 3637-3649. MR 2218992 (2007b:17048)
  • 12. C. Martínez and E. Zelmanov, Representation Theory of Jordan Superalgebras I, To appear in Transactions of the AMS.
  • 13. C. Martínez and E. Zelmanov, Jordan Superalgebras and their Representations, Contemporary Mathematics 483 (2009), 179-194. MR 2497959
  • 14. N.A. Pisarenko, The structure of alternative superbimodules, Algebra and Logic 33 no.6 (1995), 386-397. MR 1347266 (96h:17042)
  • 15. M. Racine and E. Zelmanov, Simple Jordan superalgebras with semisimple even part, J. of Algebra 270 no.2 (2003), 374-444. MR 2019625 (2005b:17063)
  • 16. I. P. Shestakov, Prime alternative superalgebras of arbitrary characteristic, Algebra and Logic 36 no.6 (1997), 389-420. MR 1657313 (99k:17006)
  • 17. A.S. Shtern, Representation of an exceptional Jordan superalgebra, Funktsional Anal. i Prilozhen 21 (1987), 93-94. MR 911787 (89b:17009)
  • 18. A.S. Shtern, Representations of finite-dimensional Jordan superalgebras of Poisson brackets, Comm. in Algebra 23 no.5 (1995), 1815-1823. MR 1323702 (96e:17069)
  • 19. C.T.C. Wall, Graded Brauer groups, J. Reine Angew Math. 213 (1964), 187-199. MR 0167498 (29:4771)
  • 20. K. A. Zhevlakov, A. M. Slinko, I. P. Shestakov and A. I. Shirshov, Rings that are rearly associative, Academic Press, New York, 1982. MR 668355 (83i:17001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 17C70, 17C55, 17B10, 17B60

Retrieve articles in all journals with MSC (2000): 17C70, 17C55, 17B10, 17B60


Additional Information

Consuelo Martínez
Affiliation: Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo, s/n, 33007 Oviedo, Spain
Email: cmartinez@uniovi.es

Ivan Shestakov
Affiliation: Instituto de Matemática e Estadística, Universidade de São Paulo, Caixa Postal 66281,CEP 05315-970, São Paulo, Brasil
Email: shestak@ime.usp.br

Efim Zelmanov
Affiliation: Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112 – and – Korea Institute for Advanced Study, Seoul 130-012, Korea
Email: ezelmano@maths.ucsd.edu

DOI: https://doi.org/10.1090/S0002-9947-09-04997-6
Keywords: Jordan superalgebra, Jordan bimodule
Received by editor(s): February 27, 2008
Published electronically: November 13, 2009
Additional Notes: The first author was partially supported by MTM 2007-067884-C04-01 and FICYT IB-08-147. The author also thanks KIAS for their hospitality.
The second author was partially supported by the CNPq grant 304991/2006-6 and the FAPESP grants 05/60337-2, 05/60142-7. The author also thanks Oviedo University for their hospitality.
The third author was partially supported by the NSF
Article copyright: © Copyright 2009 American Mathematical Society

American Mathematical Society