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Journal of Lie Theory 25 (2015), No. 3, 753--774
Copyright Heldermann Verlag 2015

Lie Semigroups, Homotopy, and Global Extensions of Local Homomorphisms

Eyüp Kizil
Instituto de Ciências Matemáticas, Universidade de São Paulo, Cx. Postal 668 -- CEP 13.560-970, São Carlos -- SP, Brasil

Jimmie Lawson
Dept. of Mathematics, Louisiana State University, Baton Rouge, LA 70803, U.S.A.


\def\g{{\frak g}} For a finite dimensional connected Lie group $G$ with Lie algebra $\g$, we consider a Lie-generating Lie wedge ${\bf W}\subseteq \g$. If $S$ is a Lie subsemigroup of $G$ with subtangent wedge ${\bf W}$ we give sufficient conditions for $S$ to be free on small enough local semigroups $U\cap S$ in the sense that continuous local homomorphisms extend to global ones on $S$. The constructions involve developing a homotopy theory of $U\cap S$-directed paths. We also consider settings where the free construction leads to a simply connected covering of $S$.

Keywords: Lie semigroup, local semigroup, Lie wedge, Lie group, homotopic paths, covering semigroups.

MSC: 22A15, 22E15

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