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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

MSC:

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
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References:

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