On the Representation Theory of Lie Triple Systems

Hodge, Terrell; Parshall, Brian

doi:10.1090/S0002-9947-02-03050-7

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