PBWD bases and shuffle algebra realizations for $$U_{\varvec{v}}(L\mathfrak {sl}_n), U_{{\varvec{v}}_1,{\varvec{v}}_2}(L\mathfrak {sl}_n), U_{\varvec{v}}(L\mathfrak {sl}(m|n))$$ and their integral forms

Abstract

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.