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We define an integral form of shifted quantum affine algebras of type $A$ andconstruct Poincar\'e-Birkhoff-Witt-Drinfeld bases for them. When the shift istrivial, our integral form coincides with the RTT integral form. We prove thatthese integral forms are closed with respect to the coproduct and shifthomomorphisms. We prove that the homomorphism from our integral form to thecorresponding quantized $K$-theoretic Coulomb branch of a quiver gauge theoryis always surjective. In one particular case we identify this Coulomb branchwith the extended quantum universal enveloping algebra of type $A$. Finally, weobtain the rational (homological) analogues of the above results (provedearlier in arXiv:1611.06775, arXiv:1806.07519 via different techniques).

Shifted quantum affine algebras: integral forms in type $A$ (with appendices by Alexander Tsymbaliuk and Alex Weekes)