On the Representation Theory of Lie Triple Systems
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- by Terrell L. Hodge and Brian J. Parshall PDF
- Trans. Amer. Math. Soc. 354 (2002), 4359-4391 Request permission
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
- A. A. Beĭlinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Analysis and topology on singular spaces, I (Luminy, 1981) Astérisque, vol. 100, Soc. Math. France, Paris, 1982, pp. 5–171 (French). MR 751966
- D. J. Benson, Representations and cohomology. I, Cambridge Studies in Advanced Mathematics, vol. 30, Cambridge University Press, Cambridge, 1991. Basic representation theory of finite groups and associative algebras. MR 1110581
- A. Borel and T. A. Springer, Rationality properties of linear algebraic groups. II, Tohoku Math. J. (2) 20 (1968), 443–497. MR 244259, DOI 10.2748/tmj/1178243073
- J. M. Casas, J.-L. Loday, and T. Pirashvili, Leibniz $n$-Algebras, Forum Math. 14 (2002), 189–207.
- Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeurs, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
- Carl Faith, Algebra: rings, modules and categories. I, Die Grundlehren der mathematischen Wissenschaften, Band 190, Springer-Verlag, New York-Heidelberg, 1973. MR 0366960, DOI 10.1007/978-3-642-80634-6
- Eric M. Friedlander and Brian J. Parshall, Cohomology of Lie algebras and algebraic groups, Amer. J. Math. 108 (1986), no. 1, 235–253 (1986). MR 821318, DOI 10.2307/2374473
- Eric M. Friedlander and Brian J. Parshall, Geometry of $p$-unipotent Lie algebras, J. Algebra 109 (1987), no. 1, 25–45. MR 898334, DOI 10.1016/0021-8693(87)90161-X
- Eric M. Friedlander and Brian J. Parshall, Support varieties for restricted Lie algebras, Invent. Math. 86 (1986), no. 3, 553–562. MR 860682, DOI 10.1007/BF01389268
- Eric M. Friedlander and Brian J. Parshall, Modular representation theory of Lie algebras, Amer. J. Math. 110 (1988), no. 6, 1055–1093. MR 970120, DOI 10.2307/2374686
- Werner Greub, Stephen Halperin, and Ray Vanstone, Connections, curvature, and cohomology, Pure and Applied Mathematics, Vol. 47-III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Volume III: Cohomology of principal bundles and homogeneous spaces. MR 0400275
- Bruno Harris, Cohomology of Lie triple systems and Lie algebras with involution, Trans. Amer. Math. Soc. 98 (1961), 148–162. MR 120313, DOI 10.1090/S0002-9947-1961-0120313-0
- P. J. Hilton and U. Stammbach, A course in homological algebra, 2nd ed., Graduate Texts in Mathematics, vol. 4, Springer-Verlag, New York, 1997. MR 1438546, DOI 10.1007/978-1-4419-8566-8
- T. L. Hodge, Lie Triple Systems, Restricted Lie Triple Systems, and Algebraic Groups, Jour. Algebra 244 (2001), 533-580.
- Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
- Charles Hopkins, Rings with minimal condition for left ideals, Ann. of Math. (2) 40 (1939), 712–730. MR 12, DOI 10.2307/1968951
- Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
- Noriaki Kamiya and Susumu Okubo, On triple systems and Yang-Baxter equations, Proceedings of the Seventh International Colloquium on Differential Equations (Plovdiv, 1996) VSP, Utrecht, 1997, pp. 189–196. MR 1465993
- C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
- Ottmar Loos, Symmetric spaces. I: General theory, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0239005
- Saunders MacLane, Categories for the working mathematician, Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York-Berlin, 1971. MR 0354798
- G. P. Nagy, Algebraische kommutative Moufang-Loops (prefinal version), Doctoral Dissertation, Friedrich-Alexander-Universität Erlangen-Nürnberg, 2000.
- D. Nakano, B. Parshall and D. Vella, Support varieties for algebraic groups, Jour. reine angew. Math. 547 (2002), 15–49.
- Daniel G. Quillen, Homotopical algebra, Lecture Notes in Mathematics, No. 43, Springer-Verlag, Berlin-New York, 1967. MR 0223432, DOI 10.1007/BFb0097438
- R. W. Richardson, Orbits, invariants, and representations associated to involutions of reductive groups, Invent. Math. 66 (1982), no. 2, 287–312. MR 656625, DOI 10.1007/BF01389396
- Jonathan D. H. Smith, Representation theory of infinite groups and finite quasigroups, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 101, Presses de l’Université de Montréal, Montreal, QC, 1986. MR 859373
- Eric M. Friedlander and Andrei Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270. MR 1427618, DOI 10.1007/s002220050119
- Eric M. Friedlander and Andrei Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270. MR 1427618, DOI 10.1007/s002220050119
- Kiyosi Yamaguti, On the cohomology space of Lie triple system, Kumamoto J. Sci. Ser. A 5 (1960), 44–52 (1960). MR 132770
- Kiyosi Yamaguti, On weak representations of Lie triple systems, Kumamoto J. Sci. Ser. A 8 (1967/69), 107–114. MR 254114
Additional Information
- Terrell L. Hodge
- Affiliation: Department of Mathematics, Western Michigan University, Kalamazoo, Michigan 49008
- Email: terrell.hodge@wmich.edu
- Brian J. Parshall
- Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
- MR Author ID: 136395
- Email: bjp8w@virginia.edu
- 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.
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 4359-4391
- MSC (2000): Primary 17B99; Secondary 18G60, 20G05
- DOI: https://doi.org/10.1090/S0002-9947-02-03050-7
- MathSciNet review: 1926880