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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

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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.

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Notes

  1. It should be noted right away that these forms can be defined over \({\mathbb {Z}}[{\varvec{v}},{\varvec{v}}^{-1}]\) and \({\mathbb {Z}}[{\varvec{v}}_1,{\varvec{v}}_2,{\varvec{v}}^{-1}_1,{\varvec{v}}^{-1}_2]\), respectively, and all our results for the integral forms generalize verbatim to this setting as well.

  2. These are trigonometric counterparts of the elliptic shuffle algebras of Feigin–Odesskii [12,13,14].

  3. Following [12,13,14] the role of the wheel conditions is exactly to replace complicated Serre relations.

  4. In the formal setup (when working over \({\mathbb {C}}[[\hbar ]]\) rather than over \({\mathbb {C}}({\varvec{v}})\)), this goes back to [10, Corollary 1.4].

  5. One of the benefits of our proof is that it will be directly generalized to establish the isomorphisms of Theorems 4.10 and 5.18 below, for which no analogue of [31, Theorem 1.1] is known at the moment.

  6. As pointed out in the introduction, the linear independence can be deduced from the general arguments based on the flatness of the deformation and the PBW property of \(U(\mathfrak {sl}_n[t,t^{-1}])\). However, the specialization maps of (3.7) and formulas (3.18, 3.19) will be used below to prove that \(\{e_h\}_{h\in H}\) span \(U^>_{\varvec{v}}(L\mathfrak {sl}_n)\). We will use the same approach for two-parameter quantum loop algebra, for which the general arguments do not apply.

  7. This can be checked by treating each of the following cases separately: \(j=j'=i=i'\), \(j=j'=i<i'\), \(j=j'<i=i'\), \(j=j'<i<i'\), \(j<j'\le i'<i\), \(j<j'<i'=i\), \(j<j'<i<i'\), \(j=i<j'=i'\), \(j=i<j'<i'\), \(j<j'=i=i'\), \(j<j'=i<i'\), \(j<i<j'\le i'\), where we set \(j:=j(\beta ),j':=j(\beta '),i:=i(\beta ),i':=i(\beta ')\).

  8. To be more precise, this recovers the algebra of loc.cit. with the trivial central charges.

  9. To be more precise, one actually needs to use the classical Lie superalgebra \(A(m-1,n-1)\) in place of \(\mathfrak {sl}(m|n)\), which do coincide when \(m\ne n\). However, we shall ignore this difference, since we will be working only with the positive subalgebras and those are isomorphic: \(U^>_{\varvec{v}}(L\mathfrak {sl}(m|n))\simeq U^>_{\varvec{v}}(LA(m-1,n-1))\).

  10. See [17, Appendix B] for a correction of a gap in the proof of [28].

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Acknowledgements

I am indebted to Pavel Etingof, Boris Feigin, Michael Finkelberg, and Andrei Neguţ for numerous stimulating discussions over the years; to Naihuan Jing for a useful correspondence on two-parameter quantum algebras; to Luan Bezerra and Evgeny Mukhin for a useful correspondence on the quantum affine superalgebras; to anonymous referees for extremely useful suggestions which significantly improved the overall exposition. I am also grateful to MPIM (Bonn, Germany), IPMU (Kashiwa, Japan), and RIMS (Kyoto, Japan) for the hospitality and wonderful working conditions in the summer 2018 when the first stages of this project were performed. I would like to thank Tomoyuki Arakawa and Todor Milanov for their invitations to RIMS and IPMU, respectively. I gratefully acknowledge NSF Grants DMS-1821185, DMS-2001247, and DMS-2037602.

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    Alexander Tsymbaliuk

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Tsymbaliuk, A. 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. Sel. Math. New Ser. 27, 35 (2021). https://doi.org/10.1007/s00029-021-00634-5

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