Old and new on Sl(2). (English) Zbl 0605.22004

The present paper originated in the authors efforts to arrive at a thorough understanding of what a Lie subsemigroup of Sl(2, \({\mathbb{R}})\), or its simply connected covering group \((2, {\mathbb{R}})^{\sim}\) looks like. To this end the authors first provide a detailed computational and geometrical survey over the Lie algebra \({\mathfrak sl}(2, {\mathbb{R}})\). Together with well-known material, this survey also contains a new global parametrization of \((2, {\mathbb{R}})^{\sim}\), in terms of which the authors explicitly compute the exponential function, the logarithm and its maximal domain, and similar devices.
The two-dimensional subalgebras of \({\mathfrak sl}(2, {\mathbb{R}})\) are characterized as the tangent planes of the cone \(Kill(X,X)=0\). After these preparations, the authors determine the conjugacy classes of the Lie subsemialgebras of \({\mathfrak sl}(2, {\mathbb{R}})\) and observe that a wedge in \({\mathfrak sl}(2, {\mathbb{R}})\) is a Lie wedge iff it is either a proper cone or a Lie semialgebra. Based on the infinitesimal theory thus developed the authors then classify the infinitesimally generated subsemigroups of Sl(2, \({\mathbb{R}})\) and of \((2, {\mathbb{R}})^{\sim}\).
Reviewer: W.Ruppert


22A20 Analysis on topological semigroups
22E60 Lie algebras of Lie groups
22E46 Semisimple Lie groups and their representations
22A15 Structure of topological semigroups
17B99 Lie algebras and Lie superalgebras
Full Text: DOI EuDML


[1] Birkhoff, G., Lattice theory, 4th.Ed., Providence 1973
[2] Bourbaki, N., Groupes et algèbres de Lie, Chap 2-3, Paris, Hermann, 1972 · Zbl 0244.22007
[3] Brockett, R.W., Lie algebras and Lie groups in control theory, in: Geometric methods in systems theory, Hingham, Reidel 1973, 43-82
[4] Dieudonné, J., La géometrie des groupes classiques, Berlin-Heidelberg-New York, Springer 1963 · Zbl 0111.03102
[5] Dobbins, J.G., Well-bounded semigroups in locally compact groups, Math.Z.148(1976), 155-167 · Zbl 0321.22010
[6] Fuchs, L., Partially ordered algebraic systems, Oxford, London, New York, Pergamon Press 1963 · Zbl 0137.02001
[7] Graham, G., Differential semigroups, Lecture Notes in Math.998 (1983), 57-127
[8] Helgason, S., Differential geometry and symmetric spaces, New York, Acad.Press 1962 · Zbl 0111.18101
[9] Hilgert, J. and K.H. Hofmann, The Sl(2)-Handbook I, II and III, Notes of the Seminar Lie Theory of Semigroups S1S, 05-27-83, 06-14-83 and 10-01-83, respectively
[10] ?, Semigroups in Lie groups, Lie semi-algebras in Lie algebras, Trans. Amer. Math. Soc.288 (1985), 481-504 · Zbl 0565.22007
[11] ?, Lie semialgebras are real phenomena, Math. Ann.270 (1985), 97-103 · Zbl 0543.17001
[12] - The invariance of cones and wedges under flows, Preprint Nr.796, Technische Hochschule Darmstadt, Dezember 1983, Geometriae Dedicata, to appear
[13] - On Sophus Lie’s Fundamental Theorems, Preprint Nr.799, Technische Hochschule Darmstadt, Februar 1984. Journal of Functional Analysis, to appear
[14] Hofmann, K.H. and J.D. Lawson, Foundations of Lie semigroups, Lecture Notes in Math.998 (1983), 128-201 · Zbl 0524.22003
[15] ? Divisible subsemigroups of Lie groups, J.London Math. Soc. (2) 27(1983), 427-434 · Zbl 0505.22002
[16] Hofmann, K.H. and P.S. Mostert, Elements of Compact Semigroups, Ch.E. Merrill, Columbus (Ohio), 1966 · Zbl 0161.01901
[17] Lang, S., S12(lo), Reading, Addison Wesley 1975
[18] 01’shanskii, G.I., Invariant cones in Lie algebras, Lie semigroups, and the holomorphic discrete series, Funktional Analysis and Applications15(1981), 275-285 · Zbl 0503.22011
[19] - Convex cones in symmetric Lie algebras, Lie semigroups and invariant cousal (order)structures on pseudo Riemannian symmetric spaces, Dok.Soviet.Math.20(1982) · Zbl 0512.22012
[20] Paneitz, S.M., Invariant convex cones and causality in semisimple Lie algebras and groups, J.of Functional Analysis43 (1981), 313-359 · Zbl 0476.22009
[21] ? Classification of invariant convex cones in simple Lie algebras, Arkiv för Math.21 (1984), 217-228 · Zbl 0526.22016
[22] Poguntke, D., Well-bounded semigroups in connected groups, Semigroup Forum15(1977), 159-167 · Zbl 0393.22001
[23] Vinberg, E.B., Invariant cones and orderings in Lie groups, Functional Analysis and Applications14(1980), 1-13 · Zbl 0452.22014
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