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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Semigroups in Lie groups, semialgebras in Lie algebras

Authors: Joachim Hilgert and Karl H. Hofmann
Journal: Trans. Amer. Math. Soc. 288 (1985), 481-504
MSC: Primary 22E15; Secondary 22A99, 22E05
MathSciNet review: 776389
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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.

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Keywords: Local semigroups in Lie groups, convex cones in Lie algebras, wedges in $ {{\mathbf{R}}^n}$, duality of cones and wedges, Lie semialgebra and Lie algebra, tangent hyperplane tangent space
Article copyright: © Copyright 1985 American Mathematical Society

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