Convex orders in root systems and quantum tangent spaces

We classify the reduced decompositions of the longest element $w_0$ of the Weyl group of a semisimple Lie algebra $\mathfrak{g}$ for which the corresponding Lusztig quantum root vectors span a quantum tangent space in $U_q(\mathfrak{g})$. We introduce a combinatorial condition on the convex order induced by a reduced decomposition, tangency, and prove that a reduced decomposition gives rise to a quantum tangent space if and only if it is tangent. The proof relies on a description of the coproduct of quantum root vectors in terms of the convex order, and on a projection in the right weak order attached to each positive root. Up to commutation, tangent reduced decompositions are obtained by concatenating the longest minimal coset representatives along a chain of cominuscule parabolic quotients; in particular there are none in types $E_8$, $F_4$ and $G_2$.