### Representation theory seminar, The Ohio State University

**Oleksandr Tsymbaliuk, Purdue University**

*Quantum loop groups and shuffle algebras via Lyndon words*

Organizer's time: 2021-04-14 16:15 America/New_York

Duration: 1 hour

Access information: Online: via Zoom; Meeting Id: 992 7811 3905

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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 of U_q(Ln) into A, in terms of pole and wheel conditions. Joint work with Andrei Negut.

Submitted by: Sachin Gautam