PBWD bases and shuffle algebra realizations for $U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ and their integral forms

TL;DR

This paper constructs PBWD bases for various quantum loop algebras and their integral forms using shuffle algebra realizations, proving existing conjectures and extending the framework to super and two-parameter cases.

Contribution

It introduces explicit shuffle algebra realizations for quantum loop algebras, proving conjectures and extending PBWD bases to super and two-parameter quantum groups.

Findings

Proved a conjecture for super quantum loop algebras.

Constructed PBWD bases for RTT and Lusztig forms.

Provided shuffle algebra realizations for Yangians and dual subalgebras.

Abstract

We construct a family of PBWD (Poincar\'e-Birkhoff-Witt-Drinfeld) bases for the quantum loop algebras $U_v(L\mathfrak{sl}_n), U_{v_1,v_2}(L\mathfrak{sl}_n), U_v(L\mathfrak{sl}(m|n))$ in the new Drinfeld realizations. In the 2-parameter case, this proves Theorem 3.11 of [Hu-Rosso-Zhang] (stated in loc. cit. without a proof), while in the super case it proves a conjecture of [Zhang]. The main ingredient in our proofs is the interplay between those quantum loop algebras and the corresponding shuffle algebras, which are trigonometric counterparts of the elliptic shuffle algebras of Feigin-Odesskii. Our approach is similar to that of [Enriquez] in the formal setting, but the key novelty is an explicit shuffle algebra realization of the corresponding algebras, which is of independent interest. This also allows us to strengthen the above results by constructing a family of PBWD bases for the RTT forms of those quantum loop algebras as well as for the Lusztig form of $U_v(L\mathfrak{sl}_n)$. The rational counterparts provide shuffle algebra realizations of the type $A$ (super) Yangians and their Drinfeld-Gavarini dual subalgebras.