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 structure of real transitive Lie algebras


Author: Jack F. Conn
Journal: Trans. Amer. Math. Soc. 286 (1984), 1-71
MSC: Primary 58H05; Secondary 17B65, 22E65
DOI: https://doi.org/10.1090/S0002-9947-1984-0756031-0
MathSciNet review: 756031
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we examine some of the ways in which abstract algebraic objects in a transitive Lie algebra $ L$ are expressed geometrically in the action of each transitive Lie pseudogroup $ \Gamma $ associated to $ L$. We relate those chain decompositions of $ \Gamma $ which result from considering $ \Gamma $-invariant foliations to Jordan-Hölder sequences (in the sense of Cartan and Guillemin) for $ L$. Local coordinates are constructed which display the nature of the partial differential equations defining $ \Gamma $; in particular, locally homogeneous pseudocomplex structures (also called $ {\text{CR}}$-structures) are associated to the nonabelian quotients of complex type in a Jordan-Hölder sequence for $ L$.


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

  • [1] N. Bourbaki, Élements de mathématique, algèbre, Hermann, Paris, 1958.
  • [2] -, Élements de mathématique, algèbre commutative, Hermann, Paris, 1958.
  • [3] É. Cartan, Les groupes de transformations continus, infinis, simples, Ann. Sci. École Norm. Sup. 26 (1909), 93-161. MR 1509105
  • [4] J. F. Conn, A new class of counterexamples to the integrability problem, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), 2655-2658. MR 0464334 (57:4266)
  • [5] -, Non-abelian minimal closed ideals of transitive Lie algebras, Princeton Univ. Press and Univ. of Tokyo Press, Princeton, 1981. MR 595686 (82i:58077)
  • [6] C. Freifeld, The cohomology of transitive filtered modules. I. The first cohomology group, Trans. Amer. Math. Soc. 144 (1969), 475-491. MR 0283818 (44:1048)
  • [7] H. Goldschmidt, Sur la structure des équations de Lie. I. Le troisième théorème fondamental, J. Differential Geom. 6 (1972), 357-373; II. Équations formellement transitives, J. Differential Geom. 7 (1972), 67-95; III. La cohomologie de Spencer, J. Differential Geom. 11 (1976), 167-223. MR 0301768 (46:923)
  • [8] -, The integrability problem for Lie equations, Bull. Amer. Math. Soc. 84 (1978), 531-546. MR 0517116 (58:24418)
  • [9] -, On the non-linear cohomology of Lie equations. V, J. Differential Geom. 16 (1981), 595-674; VI (to appear). MR 661657 (83k:58094)
  • [10] H. Goldschmidt and D. C. Spencer, On the non-linear cohomology of Lie equations. I, II. Acta Math. 136 (1976), 103-239; III, IV. J. Differential Geom. 13 (1978), 409-526. MR 0445566 (56:3904a)
  • [11] M. Golubitsky, Primitive actions and maximal subgroups of Lie groups, J. Differential Geom. 7 (1972), 175-191. MR 0327855 (48:6197)
  • [12] V. W. Guillemin, A Jordan-Hölder decomposition for a certain class of infinite-dimensional Lie algebras, J. Differential Geom. 2 (1968), 313-345. MR 0263882 (41:8481)
  • [13] -, Infinite-dimensional primitive Lie algebras, J. Differential Geom. 4 (1970), 257-282. MR 0268233 (42:3132)
  • [14] -, Representations of Lie algebras as derivations of formal power series rings, 1969, unpublished.
  • [15] V. W. Guillemin and S. Sternberg, An algebraic model of transitive differential geometry, Bull. Amer. Math. Soc. 70 (1964), 16-47. MR 0170295 (30:533)
  • [16] N. Iwahori, On real irreducible representations of Lie algebras, Nagoya Math. J. 14 (1959), 59-83. MR 0102534 (21:1325)
  • [17] N. Jacobson, Lie algebras, Interscience, New York, 1962. MR 0143793 (26:1345)
  • [18] G. Köthe, Topologische lineare räume. I, Springer-Verlag, Berlin. 1966.
  • [19] V. V. Morozov, Sur les groupes primitifs, Mat. Sb. (N.S.) 5(47) (1939), 355-390. (Russian with French summary) MR 0001557 (1:258e)
  • [20] L. Nirenberg, A complex Frobenius theorem, Seminar on Analytic Functions, Institute for Advanced Study 1 (1957), 172-179.
  • [21] D. S. Rim, Deformation of transitive Lie algebras, Ann. of Math. (2) 83 (1966), 339-357. MR 0199315 (33:7463)
  • [22] S. Shnider, The classification of real primitive Lie algebras, J. Differential Geom. 4 (1970), 81-89. MR 0285574 (44:2792)
  • [23] I. M. Singer and S. Sternberg, The infinite groups of Lie and Cartan. Part I (The transitive groups), J. Analyse Math. 15 (1965), 1-114. MR 0217822 (36:911)
  • [24] D. C. Spencer, Deformations of structures on manifolds defined by transitive, continuous pseudogroups. I, II, Ann. of Math. (2) 76 (1962), 306-445. MR 0156363 (27:6287a)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 58H05, 17B65, 22E65

Retrieve articles in all journals with MSC: 58H05, 17B65, 22E65


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1984-0756031-0
Article copyright: © Copyright 1984 American Mathematical Society

American Mathematical Society