## Semigroups in Lie groups, semialgebras in Lie algebras

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- by Joachim Hilgert and Karl H. Hofmann
- Trans. Amer. Math. Soc.
**288**(1985), 481-504 - DOI: https://doi.org/10.1090/S0002-9947-1985-0776389-7
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## Abstract:

Consider a subsemigroup of a Lie group containing the identity and being ruled by one-parameter semigroups near the identity. We associate with it the set $W$ of its tangent vectors at the identity and obtain a subset of the Lie algebra $L$ of the group. The set $W$ has the following properties: (i) $W + W = W$, (ii) ${{\mathbf {R}}^ + } \cdot \;W \subset W$, (iii) ${W^ - } = W$, and, the crucial property, (iv) for all sufficiently small elements $x$ and $y$ in $W$ one has $x \ast y = x + y + \frac {1} {2}[x,y] + \cdots$ (Campbell-Hausdorff!) $\in W$. We call a subset $W$ of a finite-dimensional real Lie algebra $L$ a Lie semialgebra if it satisfies these conditions, and develop a theory of Lie semialgebras. In particular, we show that a subset $W$ satisfying (i)-(iii) is a Lie semialgebra if and only if, for each point $x$ of $W$ and the (appropriately defined) tangent space ${T_x}$ to $W$ in $x$, one has $[x,{T_x}] \subset {T_x}$. (The Lie semialgebra $W$ of a subgroup is always a vector space, and for vector spaces $W$ we have ${T_x} = W$ for all $x$ in $W$, and thus the condition reduces to the old property that $W$ is a Lie algebra.) In the introduction we fully discuss all Lie semialgebras of dimension not exceeding three. Our methods include a full duality theory for closed convex wedges, basic Lie group theory, and certain aspects of ordinary differential equations.## References

- George Phillip Barker,
*Faces and duality in convex cones*, Linear and Multilinear Algebra**6**(1978/79), no. 3, 161–169. MR**512991**, DOI 10.1080/03081087808817234 - N. Bourbaki,
*Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie*, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1349, Hermann, Paris, 1972. MR**0573068**
R. W. Brockett, - 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. MR**724628**, DOI 10.1007/BFb0062028 - Joachim Hilgert and Karl H. Hofmann,
*Lie theory for semigroups*, Semigroup Forum**30**(1984), no. 2, 243–251. MR**760225**, DOI 10.1007/BF02573456
—, - Ronald Hirschorn,
*Topological semigroups, sets of generators, and controllability*, Duke Math. J.**40**(1973), 937–947. MR**325837** - Karl H. Hofmann and Jimmie D. Lawson,
*The local theory of semigroups in nilpotent Lie groups*, Semigroup Forum**23**(1981), no. 4, 343–357. MR**638578**, DOI 10.1007/BF02676658 - 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. MR**724629**, DOI 10.1007/BFb0062029
—, - Karl H. Hofmann and Jimmie D. Lawson,
*Divisible subsemigroups of Lie groups*, J. London Math. Soc. (2)**27**(1983), no. 3, 427–434. MR**697136**, DOI 10.1112/jlms/s2-27.3.427
—, - 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. MR**0089373** - Velimir Jurdjevic and Héctor J. Sussmann,
*Control systems on Lie groups*, J. Differential Equations**12**(1972), 313–329. MR**331185**, DOI 10.1016/0022-0396(72)90035-6 - R. P. Langlands,
*On Lie semi-groups*, Canadian J. Math.**12**(1960), 686–693. MR**121667**, DOI 10.4153/CJM-1960-061-0 - Charles Loewner,
*On semigroups in analysis and geometry*, Bull. Amer. Math. Soc.**70**(1964), 1–15. MR**160192**, DOI 10.1090/S0002-9904-1964-11015-6 - G. I. Ol′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). MR**639200**
—, - R. Tyrrell Rockafellar,
*Convex analysis*, Princeton Mathematical Series, No. 28, Princeton University Press, Princeton, N.J., 1970. MR**0274683**
L. J. M. Rothkrantz, - È. B. Vinberg,
*Invariant convex cones and orderings in Lie groups*, Funktsional. Anal. i Prilozhen.**14**(1980), no. 1, 1–13, 96 (Russian). MR**565090**

*Lie algebras and Lie groups in control theory*, Reidel, Hingham, Mass., 1973, pp. 43-82. G. Graham, Differentiability and semigroups, Dissertation, University of Houston, 1979.

*On Sophus Lie’s fundamental theorem*, Preprint No. 799, TH Darmstadt, 1984; J. Funct. Analysis (to appear). —,

*The invariance of cones and wedges under flows*, Preprint No. 796, TH Darmstadt, 1983 (submitted).

*On Sophus Lie’s fundamental theorems*I

*and*II, Indag. Math.

**45**(1983), 453-466; ibid.

**46**(1984), 255-265.

*On the description of the closed local semigroup generated by a wedge*, SLS-Memo, 03-16-83, 1983.

*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.

*Transformatiehalfgroepen van niet-compacte hermitische symmetrische Ruimten*, Dissertation, University of Amsterdam, 1980. S. Straszewicz,

*Über exponierte Punkte abeschlossener Punktmengen*, Fund. Math.

**24**(1935), 139-143.

## Bibliographic Information

- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**288**(1985), 481-504 - MSC: Primary 22E15; Secondary 22A99, 22E05
- DOI: https://doi.org/10.1090/S0002-9947-1985-0776389-7
- MathSciNet review: 776389