Universal covering semigroups of Lie semigroups

Abstract

Monotonic homotopy was considered by Lawson in [law] and [law1], where the main purpose was to extend to Lie semigroups the classical construction of universal covering groups. Monotonic homotopy between monotonic curves of a Lie semigroup S is a variant of the usual homotopy linking continuously monotonic curves of S through monotonic curves. We are interessted in this Project for a possible identification of the semigroup \Gamma(S) of monotonic homotopy classes with the universal covering semigroup \tilde(S) of S (or a closed subsemigroup of it). This problem, which is actually a conjecture formulated by Lawson himself, was proved for a particular class of semigroups, called Ol'shanskii semigroups. We intend to deal with Lie semigroups in general using the preliminary results we have obtained in [ckl] on the covering space \Gamma(\Sigma,x), x in M, of a conic control system \Sigma since both “\Gamma(\Sigma,x) and \tilde(S) are obtained via the same aproach. There exists a good perspective that we could achieve our objective since this covering space is simply connected which would allow us to identify it with the universal covering semigroup of S. References. [ckl] Colonius, F., Kizil, E. e San Martin, L: Covering space for monotonic homotopy of trajectories of control systems, Journal of Differential Equations, Vol 216, Issue 2, pg.324-353, 2005.[law] J. Lawson: Universal Objects in Lie Semigroup Theory. In Positivity in Lie Theory (J. Hilgert, J.D. Lawson, K.-H. Neeb, E.B. Vinberg, editors) de Gruyter Expositions in Mathematics 20 (1995).[law1] J. Lawson: Free Local Semigroup Constructions. Monatsh. Math. 121 (1996), 309-333. (AU)