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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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