Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 4 (2008), 020, 10 pages      arXiv:0802.2332
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

An Infinite Dimensional Approach to the Third Fundamental Theorem of Lie

Richard D. Bourgin and Thierry P. Robart
Department of Mathematics, Howard University, Washington DC 20059, USA

Received November 02, 2007, in final form January 16, 2008; Published online February 16, 2008

We revisit the third fundamental theorem of Lie (Lie III) for finite dimensional Lie algebras in the context of infinite dimensional matrices.

Key words: Lie algebra; Ado theorem; integration; Lie group; infinite dimensional matrix; representation.

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  1. Ado I.D., Note on the representation of finite continuous groups by means of linear substitutions, Izv. Fiz.-Mat. Obsch. (Kazan) 7 (1935), 1-43 (in Russian).
  2. Ado I.D., The representation of Lie algebras by matrices, Transl. Amer. Math. Soc. (1) 9 (1962), 308-327.
  3. Beltitua D., Neeb K.-H., Finite-dimensional Lie subalgebras of algebras with continuous inversion, math.FA/0603420.
  4. Bochner S., Formal Lie groups, Ann. of Math. 47 (1946), 192-201.
  5. Bourgin R., Robart T., Generalization of the concept of invertibility for infinite dimensional matrices, submitted.
  6. Bourgin R., Robart T., On Ado's theorem, in preparation.
  7. Cartan É., La Topologie des Groupes de Lie, Exposés de Géométrie No. 8, Hermann, Paris, 1936.
  8. Grauert H., On Levi's problem and the imbedding of real-analytic manifolds, Ann. of Math. 68 (1958), 460-472.
  9. Henrici P., Applied and computational complex analysis, Vol. 1, Wiley, 1974.
  10. Malcev A.I., Sur les groupes topologiques locaux et complets, C. R. Acad. Sci. URSS 32 (1941), 606-608.
  11. Morrey C.B., The analytic embedding of abstract real-analytic manifolds, Ann. of Math. 68 (1958), 159-201.
  12. Olver P.J., Non-associative local Lie groups, J. Lie Theory 6 (1996), 23-51.
  13. Robart T., About the local and formal geometry of PDE, Contemp. Math. 285, (2001), 183-194.
  14. Smith P.A., Topological foundations in the theory of continuous transformation groups, Duke Math. J. 2 (1936), 246-279.
  15. Smith P.A., Topological groups, in Proceedings of the International Congress of Mathematicians (1950, Cambridge, Mass.), Vol. 2, Amer. Math. Soc., Providence, R.I., 1952, 436-441.
  16. Weinstein A., Groupoids: unifying internal and external symmetry, Notices Amer. Math. Soc. 43 (1996), 744-752, math.RT/9602220.

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