## 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

### Keywords:

Lie semigroups; Sl(2, $${bbfR})$$; wedge; Lie semialgebra
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### References:

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