We construct a family of PBWD (Poincaré–Birkhoff–Witt–Drinfeld) bases for the quantum loop algebras \(U_{\varvec{v}}(L\mathfrak {sl}_n),U_{{\varvec{v}}_1,{\varvec{v}}_2}(L\mathfrak {sl}_n),U_{\varvec{v}}(L\mathfrak {sl}(m|n))\) in the new Drinfeld realizations. In the 2-parameter case, this proves (Hu et al. in Commun Math Phys 278(2):453–486, 2008, Theorem 3.11) (stated in loc. cit. without a proof), while in the super case it proves a conjecture of Zhang (Math. Z. 278(3–4):663–703, 2014). 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 and Odesskii (Anal. i Prilozhen 23(3):45–54, 1989; Anal i Prilozhen 31(3):57–70, 1997; Integrable Structures of Exactly Solvable Two-Dimensional Models of Quantum Field Theory (Kiev, 2000). NATO Sci Ser II Math Phys Chem, vol 35, Kluwer Academic Publishers, Dordrecht, pp 123–137, 2001). Our approach is similar to that of Enriquez (J Lie Theory 13(1):21–64, 2003) 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 \(U_{\varvec{v}}(L\mathfrak {sl}_n)\). The rational counterparts provide shuffle algebra realizations of type A (super) Yangians and their Drinfeld–Gavarini dual subalgebras.

我们为量子环路代数\（U _ {\ varvec {v}}（L \ mathfrak {sl} _n），U _ {{\ varvec {v}}在新的Drinfeld实现中，_1，{\ varvec {v}} _ 2}（L \ mathfrak {sl} _n），U _ {\ varvec {v}}（L \ mathfrak {sl}（m | n））\）。在2参数的情况下，这证明了（Hu等人，Commun Math Phys 278（2）：453-486，2008，定理3.11）（在上述引证中有述）  。没有证明），而在超级案例中，它证明了Zhang的猜想（Math。Z. 278（3–4）：663–703，2014）。我们的证明中的主要成分是那些量子环代数和相应的随机代数之间的相互作用，它们是Feigin和Odesskii的椭圆随机代数的三角对应物（Anal。i Prilozhen 23（3）：45-54，1989; Anal i Prilozhen 31（3）：57-70，1997年；完全可解决的二维量子场论模型的可积结构（基辅，2000年），《北约科学第二期数学物理化学》，第35卷，克鲁沃学术出版社，多德雷赫特，第pp 123–137，2001年）。在形式上，我们的方法与Enriquez（J Lie Theory 13（1）：21–64，2003）相似，但是关键的新颖之处是对相应代数的显式改组代数实现，这是独立感兴趣的。\（U _ {\ varvec {v}}（L \ mathfrak {sl} _n）\）。有理对应物提供了A型（超级）洋基人及其Drinfeld-Gavarini对子子代的混洗代数实现。