On existence of invariant measures
Tóm tắt
Let G be a Lie group, H ≤ G a closed subgroup and M ≈ G/H. In [14] André Weil gave a necessary and sufficient condition for the existence of invariant measures on homogeneous spaces of arbitrary locally compact groups. For Lie groups using the structure theory we give a neater necessary and sufficient condition for the existence of a G-invariant measure on M, cf. Theorems (2.1) and (3.2) in the introduction.
Tài liệu tham khảo
N. Bourbaki, Lie groups and Lie algebras, translated from the French original by Andrew Pressley, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002.
Shiing-shen Chern, On integral geometry in Klein spaces. Ann. of Math., 43(2), (1942), 178–189.
A. Gaal Steven, Linear analysis and representation theory. Die Grundlehren der mathematischenWissenschaften, Band 198. Springer-Verlag, New York-Heidelberg, 1973.
Alfred Haar, Der Massbegriff in der Theorie der kontinuierlichen Gruppen, (German) Ann. of Math., 34(1) (1933), 147–169.
Harish-Chandra, On the radical of a Lie algebra, Proc. Amer. Math. Soc., 1 (1950), 14–17.
S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press (1962).
N. Jacobson, Lie Algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, New York-London 1962.
A.W. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, Birkhäuser.
S. Kumaresan, A Course in Differential Geometry and Lie Groups, TRIM Series, Hindustan Book Agency.
Leopoldo Nachbin, The Haar Integral, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London 1965.
John von Neumann, The Uniqueness of Haar’s Measure, Mat. Sbornik, N.S., 1 (1934), 106–114.
John von Neumann, um Haarschen masz in topologischen Gruppen, Comp. Math., 1 (1934), 106–114.
G. P. Scott, The Geometries of 3-manifolds, Bull. Lond. Math Soc., 15 (1983) 401–487.
André Weil, L’intégration dans les groupes topologiques et ses applications, Actualités Scientifiques et Industrielles, Hermann, 1940.
André Weil, Sur les groupes topologiques et les groupes mesurés, C.R. Acad. Sci. Paris 202, (1936) 1147–1149.