  Multiple Schramm-Loewner Evolutions (SLE) are conformally invariant random
processes of several curves, whose construction by growth processes relies on
partition functions: M\"obius covariant solutions to a system of second order
partial differential equations. In this article, we use a quantum group
technique to construct a distinguished basis of solutions, which conjecturally
correspond to the extremal points of the convex set of probability measures of
multiple SLEs.
