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)



On the Representation Theory of Lie Triple Systems

Authors: Terrell L. Hodge and Brian J. Parshall
Journal: Trans. Amer. Math. Soc. 354 (2002), 4359-4391
MSC (2000): Primary 17B99; Secondary 18G60, 20G05
Published electronically: July 8, 2002
MathSciNet review: 1926880
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we take a new look at the representation theory of Lie triple systems. We consider both ordinary Lie triple systems and restricted Lie triple systems in the sense of [14]. In a final section, we begin a study of the cohomology of Lie triple systems.

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

  • 1. A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analyse et topologie sur les espaces singulières, I (Luminy, 1981), Astérisque 100, 1982, pp. 5-171. MR 86g:32015
  • 2. D. J. Benson, Representations and Cohomology I, Cambridge University Press, Cambridge, 1991. MR 92m:20005
  • 3. A. Borel and T. A. Springer, Rationality properties of linear algebraic groups, II, Tôhoku Math. Jour. 20 (1968), 443-497. MR 39:5576
  • 4. J. M. Casas, J.-L. Loday, and T. Pirashvili, Leibniz $n$-Algebras, Forum Math. 14 (2002), 189-207.
  • 5. M. Demazure and P. Gabriel, Groupes algébriques, Tome I: Géométrie algébrique, généralités, groupes commutatifs, North-Holland, Amsterdam, 1970. MR 46:1800
  • 6. C. Faith, Algebra: Rings, modules and categories I, Springer-Verlag, New York, 1973. MR 51:3206
  • 7. E. Friedlander and B. Parshall, Cohomology of Lie Algebras and Algebraic Groups, Amer. Jour. Math. 108 (1986), 235-253. MR 87c:20080
  • 8. E. Friedlander and B. Parshall, Geometry of $p$-Unipotent Lie Algebras, Jour. Algebra 109 (1987), 25-45. MR 89a:17017
  • 9. E. Friedlander and B. Parshall, Support varieties for restricted Lie algebras, Invent. Math. 86 (1986), 553-563. MR 88f:17018
  • 10. E. Friedlander and B. Parshall, Modular representation theory of Lie algebras, Amer. Jour. Math. 110 (1988), 1055-1094. MR 89j:17015
  • 11. W. Greub, S. Halperin, and R. Vanstone, Connections, Curvature and Cohomology, III, Academic Press, New York, 1976. MR 53:4110
  • 12. B. Harris, Cohomology of Lie Triple Systems and Lie Algebras with Involution, Trans. Amer. Math. Soc. 98 (1961), 148-162. MR 22:11068
  • 13. P. Hilton and U. Stammbach, A Course in Homological Algebra, Springer-Verlag, New York, 1997. MR 97k:18001
  • 14. T. L. Hodge, Lie Triple Systems, Restricted Lie Triple Systems, and Algebraic Groups, Jour. Algebra 244 (2001), 533-580.
  • 15. N. Jacobson, Lie and Jordan Triple Systems, Amer. Jour. Math. 71 (1949), 149-170. MR 10:426c
  • 16. N. Jacobson, General Representation Theory of Jordan Algebras, Trans. Amer. Math. Soc. 70 (1951), 509-530. MR 12:797d
  • 17. N. Jacobson, Lie Algebras, Dover, New York, 1962. MR 26:1345
  • 18. N. Kamiya and S. Okubo, On triple systems and Yang-Baxter equations, Proceedings of the Seventh International Colloquium on Differential Equations (Plovdiv, 1996), 189-196, VSP, Utrecht, 1997. MR 98m:17007
  • 19. W. G. Lister, A Structure Theory of Lie Triple Systems, Trans. Amer. Math. Soc. 72 (1952), 217-242. MR 13:619d
  • 20. O. Loos, Symmetric Spaces, vol. I, W. A. Benjamin, New York, 1969. MR 39:365a
  • 21. S. Mac Lane, Categories for the Working Mathematician, Springer-Verlag, 1971. MR 50:7275
  • 22. G. P. Nagy, Algebraische kommutative Moufang-Loops (prefinal version), Doctoral Dissertation, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2000.
  • 23. D. Nakano, B. Parshall and D. Vella, Support varieties for algebraic groups, Jour. reine angew. Math. 547 (2002), 15-49.
  • 24. D. Quillen, Homotopical algebra, Springer-Verlag Lectures Notes in Mathematics, vol. 43, New York, 1967. MR 36:6480
  • 25. R. Richardson, Orbits, invariants, and representations associated to involutions of reductive groups, Invent. Math. 66 (1982), 287-312. MR 83i:14042
  • 26. J. D. H. Smith, Representation theory of infinite groups and finite quasigroups, Séminaire de mathématiques supérieures, Université de Montréal, Montréal, 1986. MR 88a:20088
  • 27. A.A. Suslin, E.M. Friedlander, C.P. Bendel, Infinitesimal one-parameter subgroups and cohomology, Jour. Amer. Math. Soc., 10 (1997), 693-728. MR 98h:14055b
  • 28. A.A. Suslin, E.M. Friedlander, C.P. Bendel, Support varieties for infinitesimal group schemes, Jour. Amer. Math. Soc., 10 (1997), 729-759. MR 98h:14055c
  • 29. K. Yamaguti, On the cohomology space of Lie triple systems, Kumamoto J. Sci. Ser. A, 5 (1960), 44-52. MR 24:A2606
  • 30. K. Yamaguti, On Weak Representations of Lie Triple Systems, Kumamoto J. Sci., Ser. A. 8 (1968), 107-114. MR 40:7324

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 17B99, 18G60, 20G05

Retrieve articles in all journals with MSC (2000): 17B99, 18G60, 20G05

Additional Information

Terrell L. Hodge
Affiliation: Department of Mathematics, Western Michigan University, Kalamazoo, Michigan 49008

Brian J. Parshall
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903

Keywords: Lie triple systems, restricted Lie triple systems, cohomology, representation theory
Received by editor(s): October 1, 2001
Received by editor(s) in revised form: March 18, 2002
Published electronically: July 8, 2002
Additional Notes: Research supported in part by the National Science Foundation and a Research Development Award from Western Michigan University.
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society