×

Semigroups in Lie groups, semialgebras in Lie algebras. (English) Zbl 0565.22007

This paper is a thorough study of the notion of Lie semialgebras and of their analytic geometry. A Lie semialgebra is a wedge (closed convex cone) W of a finite dimensional Lie algebra L which is locally closed under the local Campbell-Hausdorff multiplication * in L.
Lie semialgebras arise as the tangent objects of local subsemigroups of Lie groups which are generated by local one-parameter semigroups. (For this and a general introduction into the foundations of Lie semigroups, see the article by K. H. Hofmann and J. D. Lawson [Proc. Conf., Oberwolfach 1981, Lect. Notes Math. 988, 128-201 (1983; Zbl 0524.22003)]. For example, the wedges which are invariant under the adjoint representation of L are of this kind, by results of G. I. Ol’shanskij [Funct. Anal. Appl. 15, 275-285 (1981; Zbl 0503.22011), Sov. Math., Dokl. 26, 97-101 (1982; Zbl 0512.22012)].
The eventual aim of the paper is to show that the given definition of a Lie semialgebra does not depend on a specific Campbell-Hausdorff neighborhood, and that the Lie semialgebras are characterized among the wedges in L by the property that \([x,T_ x]\subseteq T_ x\) for all \(x\in W\) allowing a unique tangent hyperplane \(T_ x\). Several interesting applications of this result are also presented. One of them is a direct proof of the mentioned result of Ol’shanskii; another one is the result that in an exponential Lie algebra L (in which there is a global analytic extension of the Campbell-Hausdorff multiplication *) a Lie semialgebra \(W\subseteq L\) is globally closed under *.
In § 1, along with the technical preparations for the geometric ingredients of the theory of Lie semialgebras, the authors develop in detail a complete duality theory for wedges in (not necessarily finite- dimensional) locally convex topological vector spaces. This theory contains a wealth of information of independent interest; a central result of it is a Galois correspondence between the exposed faces of a wedge W and the exposed faces of the dual wedge \(W^*\). - The duality is then combined with the notion of the tangent space of W at a point x (in an infinitesimal sense), consisting of all two-sided tangent vectors of W in X, i.e. vectors v for which there are sequences \(x_ n^+,x_ n^- \) in W such that \(\pm v=n\cdot (x_ n^{\pm}-x).\) The tangent hyperplanes (i.e. hyperplanes which are tangent spaces) play a special role; they can be characterized geometrically as unique support hyperplanes through duality by relating them to exposed faces of \(W^*.\)
In § 2, the main theorems, which we have already mentioned, are then derived. Together with the geometric results of § 1 they also imply that for a Lie semialgebra \(W\subseteq L\) and a tangent hyperplane T the subvectorspace generated by \(T\cap W\) is a Lie subalgebra in L. T itself need not be a Lie subalgebra. Only in low dimensions (dim \(L\leq 3)\) this is also true. This fact is used already in the introduction for a classification of all Lie semialgebras of dimension \(\leq 3.\)
On the other hand, § 3 describes a construction which yields examples to the contrary. If L is a Lie algebra with invariant compatible norm (e.g. a compact Lie algebra), then it is shown as a consequence of the results of this paper that \(W=\{(x,r);\| x\| \leq r\}\) is a Lie semialgebra in \(L\times {\mathbb{R}}\) which generates \(L\times {\mathbb{R}}\) as a vector space. If L is compact semisimple, no tangent space of W is a subalgebra. The construction shows, in particular, that in a compact Lie algebra with nontrivial center one can always find generating Lie semialgebras. In contrast, Lie semialgebras in compact semisimple Lie algebras are never generating (by a result of the authors to be published in Geometriae Dedicata).
\(\{\) In the last sentence, the word ”semisimple” has been inadvertently omitted in the original text of the paper, creating a contradictory statement. This misprint has been communicated to the reviewer by the second author, together with the following: In Theorem 1.17, statement (iv), the right hand side of the given description of \(T_ x^{\perp}\) must be replaced by its topological closure in case L is infinite dimensional. In Theorem 1.20, statement (2) should read \(\Phi \in EXP(W^*)\). In Lemma 1.24, \(0\neq \omega \in x^{\perp}W^*\). A bibliographical indication: the papers [HH1], [HH2] and [HH3] will appear in Manuscripta Math., J. Funct. Analysis and Geometriae Dedicata, resp.\(\}\).
Reviewer: H.Hähl

MSC:

22E05 Local Lie groups
22E60 Lie algebras of Lie groups
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] George Phillip Barker, Faces and duality in convex cones, Linear and Multilinear Algebra 6 (1978/79), no. 3, 161 – 169. · Zbl 0397.15013 · doi:10.1080/03081087808817234
[2] Группы и алгебры Ли., Издат. ”Мир”, Мосцощ, 1976 (Руссиан). Алгебры Ли, свободные алгебры Ли и группы Ли. [Лие алгебрас, фрее Лие алгебрас анд Лие гроупс]; Едитед бы А. А. Кириллов анд А. И. Кострикин; Транслатед фром тхе Френч бы Ју. А. Бахтурин анд Г. И. Ол\(^{\приме}\)šанский; Ѐлементы Математики. [Елеменц оф Матхематицс].
[3] R. W. Brockett, Lie algebras and Lie groups in control theory, Reidel, Hingham, Mass., 1973, pp. 43-82.
[4] G. Graham, Differentiability and semigroups, Dissertation, University of Houston, 1979.
[5] George E. Graham, Differentiable semigroups, Recent developments in the algebraic, analytical, and topological theory of semigroups (Oberwolfach, 1981) Lecture Notes in Math., vol. 998, Springer, Berlin, 1983, pp. 57 – 127. · Zbl 0521.58005 · doi:10.1007/BFb0062028
[6] Joachim Hilgert and Karl H. Hofmann, Lie theory for semigroups, Semigroup Forum 30 (1984), no. 2, 243 – 251. · Zbl 0584.22003 · doi:10.1007/BF02573456
[7] -, On Sophus Lie’s fundamental theorem, Preprint No. 799, TH Darmstadt, 1984; J. Funct. Analysis (to appear). · Zbl 0564.22006
[8] -, The invariance of cones and wedges under flows, Preprint No. 796, TH Darmstadt, 1983 (submitted).
[9] Ronald Hirschorn, Topological semigroups, sets of generators, and controllability, Duke Math. J. 40 (1973), 937 – 947. · Zbl 0285.22001
[10] Karl H. Hofmann and Jimmie D. Lawson, The local theory of semigroups in nilpotent Lie groups, Semigroup Forum 23 (1981), no. 4, 343 – 357. · Zbl 0531.22002 · doi:10.1007/BF02676658
[11] K. H. Hofmann and J. D. Lawson, Foundations of Lie semigroups, Recent developments in the algebraic, analytical, and topological theory of semigroups (Oberwolfach, 1981) Lecture Notes in Math., vol. 998, Springer, Berlin, 1983, pp. 128 – 201. · doi:10.1007/BFb0062029
[12] -, On Sophus Lie’s fundamental theorems I and II, Indag. Math. 45 (1983), 453-466; ibid. 46 (1984), 255-265. · Zbl 0525.22004
[13] Karl H. Hofmann and Jimmie D. Lawson, Divisible subsemigroups of Lie groups, J. London Math. Soc. (2) 27 (1983), no. 3, 427 – 434. · Zbl 0494.22002 · doi:10.1112/jlms/s2-27.3.427
[14] -, On the description of the closed local semigroup generated by a wedge, SLS-Memo, 03-16-83, 1983.
[15] Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. · Zbl 0078.10004
[16] Velimir Jurdjevic and Héctor J. Sussmann, Control systems on Lie groups, J. Differential Equations 12 (1972), 313 – 329. · Zbl 0237.93027 · doi:10.1016/0022-0396(72)90035-6
[17] R. P. Langlands, On Lie semi-groups, Canad. J. Math. 12 (1960), 686 – 693. · Zbl 0105.31204 · doi:10.4153/CJM-1960-061-0
[18] Charles Loewner, On semigroups in analysis and geometry, Bull. Amer. Math. Soc. 70 (1964), 1 – 15. · Zbl 0196.23701
[19] G. I. Ol\(^{\prime}\)shanskiĭ, Invariant cones in Lie algebras, Lie semigroups and the holomorphic discrete series, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 53 – 66, 96 (Russian).
[20] -, Convex cones in symmetric Lie algebras, Lie semigroups and invariant causal (order) structures on pseudo-Riemannian symmetric spaces, Soviet. Math. Dokl. 26 (1982), 97-101. · Zbl 0512.22012
[21] R. Tyrrell Rockafellar, Convex analysis, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. · Zbl 0193.18401
[22] L. J. M. Rothkrantz, Transformatiehalfgroepen van niet-compacte hermitische symmetrische Ruimten, Dissertation, University of Amsterdam, 1980.
[23] S. Straszewicz, Über exponierte Punkte abeschlossener Punktmengen, Fund. Math. 24 (1935), 139-143. · JFM 61.0756.09
[24] È. B. Vinberg, Invariant convex cones and orderings in Lie groups, Funktsional. Anal. i Prilozhen. 14 (1980), no. 1, 1 – 13, 96 (Russian). · Zbl 0452.22014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.