## The compression semigroup of a cone is connected.(English)Zbl 1047.22005

Let $$W\subset \mathbb R^n$$ be a pointed and generating cone, $$S(W)$$ the semigroup of matrices with positive determinant leaving $$W$$ invariant. The authors show that $$S(W)$$ is path connected. This result has the following consequence: semigroups with nonempty interior in the real special linear group $$Sl(n,\mathbb R)$$ are classified into types, each type being labelled by a flag manifold. The semigroups whose type is given by the projective space $$\mathbb P^{n-1}$$ form one of the classes. It is also shown here that the semigroups in $$Sl(n,\mathbb R)$$ leaving invariant a pointed and generating cone are the only maximal ones connected in the class of $$\mathbb P^{n-1}$$.

### MSC:

 22E15 General properties and structure of real Lie groups 15A15 Determinants, permanents, traces, other special matrix functions 20M20 Semigroups of transformations, relations, partitions, etc.
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