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# Mathematics > Representation Theory

# Title: Quantum loop groups and shuffle algebras via Lyndon words

(Submitted on 22 Feb 2021 (v1), last revised 18 Oct 2021 (this version, v2))

Abstract: 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 [26,27,40] in the finite type case. As an application, we prove that Enriquez' homomorphism [10] from the positive half of the quantum loop group to the trigonometric degeneration of Feigin-Odesskii's elliptic algebra [13] associated to $\mathfrak{g}$ is an isomorphism.

## Submission history

From: Andrei Neguţ [view email]**[v1]**Mon, 22 Feb 2021 18:55:26 GMT (783kb,D)

**[v2]**Mon, 18 Oct 2021 11:23:03 GMT (786kb,D)

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