  The universal sl_2 invariant of string links has a universality property for
the colored Jones polynomial of links, and takes values in the h-adic completed
tensor powers of the quantized enveloping algebra of sl_2. In this paper, we
exhibit explicit relationships between the universal sl_2 invariant and Milnor
invariants, which are classical invariants generalizing the linking number,
providing some new topological insight into quantum invariants. More precisely,
we define a reduction of the universal sl_2 invariant, and show how it is
captured by Milnor concordance invariants. We also show how a stronger
reduction corresponds to Milnor link-homotopy invariants. As a byproduct, we
give explicit criterions for invariance under concordance and link-homotopy of
the universal sl_2 invariant, and in particular for sliceness. Our results also
provide partial constructions for the still-unknown weight system of the
universal sl_2 invariant.
