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On the envelopes of Abelian subgroups in connected Lie groups

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Abstract

An Abelian subgroupA in a Lie groupG is said to be regular if it belongs to a connected Abelian subgroupC of the groupG (thenC is called an envelope ofA). A strict envelope is a minimal element in the set of all envelopes of the subgroupA. We prove a series of assertions on the envelopes of Abelian subgroups. It is shown that the centralizer of a subgroupA inG is transitive on connected components of the space of all strict envelopes ofA. We give an application of this result to the description of reductions of completely integrable equations on a torus to equations with constant coefficients.

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References

  1. A. I. Mal'tsev, “On the theory of the Lie groups in the large,”Mat. Sb. [Math. USSR-Sb.],16 (58), No. 2, 163–190 (1945).

    MATH  Google Scholar 

  2. T. Nono, “Sur l'application exponentielle dans les groupes de Lie,”J. Sci. Hiroshima Univ.,23, No. 3, 311–324 (1960).

    MATH  MathSciNet  Google Scholar 

  3. A. L. Onishchik, “Completely integrable equations on homogeneous spaces,”Mat. Zametki [Math. Notes],9, No. 4, 365–373 (1970).

    Google Scholar 

  4. V. V. Gorbatsevich, “A generalized Lyapunov theorem on Mal'tsev manifolds,”Mat. Sb. [Math. USSR-Sb.],94 (136), No. 2, 163–177 (1974).

    MATH  MathSciNet  Google Scholar 

  5. A. Borel, “Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes,”Tohoku Math. J.,13, No. 2, 216–240 (1961).

    MATH  MathSciNet  Google Scholar 

  6. R. Steinberg, “Torsion in reductive groups,”Adv. Math.,15, No. 1, 63–92 (1975).

    MATH  MathSciNet  Google Scholar 

  7. T. A. Springer and R. Steinberg, “Conjugacy classes,” in:Seminar on Algebraic Groups and Related Finite Groups.Lecture Notes in Math., Vol. 131 (1970), pp. 167–266.

    MathSciNet  Google Scholar 

  8. A. I. Sirota and A. S. Solodovnikov, “Noncompact semisimple Lie groups,”Uspekhi Mat. Nauk [Russian Math. Surveys],18, No. 3, 87–144 (1963).

    MathSciNet  Google Scholar 

  9. B. Koopman and A. Broun, “On the covering of analytic loci by complexes,”Trans. Amer. Math. Soc.,34, No. 2, 231–251 (1931).

    Google Scholar 

  10. K. Kuratowski,Topology, Vol. 2, Academic Press, New York, and PWN, Warsaw (1968).

    Google Scholar 

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Authors and Affiliations

  1. Moscow State University of Aviation Technology, USSR

    V. V. Gorbatsevich

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  1. V. V. Gorbatsevich
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Translated fromMatematicheskie Zametki, Vol. 59, No. 2, pp. 200–210, February, 1996.

This research was partially supported by the Russian Foundation for Basic Research under grant No. 95-10123 and by the International Science Foundation under grant No. RO-4300.

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Gorbatsevich, V.V. On the envelopes of Abelian subgroups in connected Lie groups. Math Notes 59, 141–147 (1996). https://doi.org/10.1007/BF02310953

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