Classical q-shuffle algebras provide combinatorial models for the positive half Uq(𝔫) of a finite quantum group. We define a loop version of this construction, yielding a combinatorial model for the positive half Uq(L𝔫) of a quantum loop group. In particular, we construct a PBW basis of Uq(L𝔫) indexed by standard Lyndon words, generalizing the work of Lalonde-Ram, Leclerc and Rosso in the Uq(𝔫) case. We also connect this to Enriquez’s degeneration A of the elliptic algebras of Feigin-Odesskii, proving a conjecture that describes the image of the embedding Uq(L𝔫)→A in terms of pole and wheel conditions. The talk shall conclude with the shuffle interpretations of fused currents proposed by Ding-Khoroshkin.
This is based on joint works with Andrei Negut.
This video was part of the Southeastern Lie Theory Workshop XIV on quantum structure in Lie theory.
