Quantum loop groups and shuffle algebras via Lyndon words

We study PBW bases of the untwisted quantum loop group $U_q(L\mathfrak{g})$ (in the Drinfeld new presentation) using the combinatorics of loop words, by generalizing the treatment of [29,30,43] in the finite type case. As an application, we prove that Enriquez' homomorphism [11] from the positive half of the quantum loop group to the trigonometric degeneration of Feigin-Odesskii's elliptic algebra [15] associated to $\mathfrak{g}$ is an isomorphism.