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We construct a family of PBWD (Poincar\'e-Birkhoff-Witt-Drinfeld) bases forthe 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 Drinfeldrealizations. 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 caseit proves a conjecture of [Zhang]. The main ingredient in our proofs is theinterplay between those quantum loop algebras and the corresponding shufflealgebras, which are trigonometric counterparts of the elliptic shuffle algebrasof Feigin-Odesskii. Our approach is similar to that of [Enriquez] in the formalsetting, but the key novelty is an explicit shuffle algebra realization of thecorresponding algebras, which is of independent interest. This also allows usto strengthen the above results by constructing a family of PBWD bases for theRTT 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 algebrarealizations of the type $A$ (super) Yangians and their Drinfeld-Gavarini dualsubalgebras.

selecta mathematica. new series

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