**Lyndon words and quantum groups**

Classical q-shuffle algebras provide combinatorial models for the positive half U_q(n) of a finite quantum group.

We define a loop version of this construction, yielding a combinatorial model for the positive half U_q(Ln) of a quantum loop group. In particular, we construct a PBW basis of U_q(Ln) indexed by standard Lyndon words, generalizing the work of Lalonde-Ram, Leclerc and Rosso in the U_q(n) case. We also connect this to Enriquez' degeneration A of the elliptic algebras of Feigin-Odesskii, proving a conjecture that describes the image of the embedding U_q(Ln) -> A in terms of pole and wheel conditions. This is a joint work with Andrei Negut.