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PBWD bases and shuffle algebra realizations for and their integral forms
We construct a family of PBWD (Poincar\'e-Birkhoff-Witt-Drinfeld) bases for
the quantum loop algebras in the new Drinfeld
realizations. This proves conjectures of [Hu-Rosso-Zhang] and [Zhang] and
generalizes the corresponding result of [Negut]. The key ingredient in our
proofs is the interplay between these quantum affine 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. We use the latter to introduce certain integral forms of these
quantum affine algebras and construct PBWD bases for them, which is crucially
used in [Finkelberg-Tsymbaliuk] to study integral forms of type shifted
quantum affine algebras. The rational counterparts provide shuffle algebra
realizations of the type (super) Yangians. Finally, we also establish the
shuffle algebra realizations of the integral forms of [Grojnowski] and
[Chari-Pressley].Comment: v1: 14 pages, comments are welcome! v2: 16 pages; introduction
expanded, Remarks 2.18, 4.7, 5.8 added. v3: 31 pages; significantly improved,
twice expanded, and entirely rewritten version; corrected description of the
integral forms of shuffle algebras (Section 3.3); new: Sections 3.3, 3.4, 6,
7, 8; list of references expande
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