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

Abstract

We define an integral form of shifted quantum affine algebras of type $A$ and construct Poincaré-Birkhoff-Witt-Drinfeld bases for them. When the shift is trivial, our integral form coincides with the RTT integral form. We prove that these integral forms are closed with respect to the coproduct and shift homomorphisms. We prove that the homomorphism from our integral form to the corresponding quantized $K$-theoretic Coulomb branch of a quiver gauge theory is always surjective. In one particular case we identify this Coulomb branch with the extended quantum universal enveloping algebra of type $A$. Finally, we obtain the rational (homological) analogues of the above results (proved earlier in arXiv:1611.06775, arXiv:1806.07519 via different techniques).

arXiv commentv1: 65 pages. v2: 67 pages; added a dominance condition in Section 2(vii), another definition in Appendix A(viii), the injectivity of $\mathfrak{g}\to Y$ in Appendix B(ii). v3: 70 pages; table of contents added, Section 3(vi) updated and Remark 4.33 added. v4: 70 pages; this is a slight modification of the version published in the Arnold Mathematical Journal, the difference is in the introduction

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Cited by in corpus (19)

• BFN Springer Theory
• Rational Lax matrices from antidominantly shifted extended Yangians: BCD types
• Correlators on the wall and ${\mathfrak{sl}}_n$ spin chain
• Vertex Representations for Yangians of Kac-Moody algebras
• PBWD bases and shuffle algebra realizations for $U_v(L{\mathfrak{sl}}_n),U_{v_1,v_2}(L{\mathfrak{sl}}_n),U_v(L\mathfrak{sl}(m|n))$ and their integral forms
• The formal shift operator on the Yangian double
• Poles of finite-dimensional representations of Yangians
• Multiplicative slices, relativistic Toda and shifted quantum affine algebras
• The meromorphic R-matrix of the Yangian
• Duality of Lusztig and RTT integral forms of $U_v(L{\mathfrak{sl}}_n)$
• Coherent IC-sheaves on type $A_n$ affine Grassmannians and dual canonical basis of affine type $A_1$
• Braid group actions, Baxter polynomials, and affine quantum groups
• Shuffle algebras and their integral forms: specialization map approach in types $B_n$ and $G_2$
• On a conjecture of Khoroshkin and Tolstoy
• Difference operators via GKLO-type homomorphisms: shuffle approach and application to quantum Q-systems
• The R-matrix presentation for the rational form of a quantized enveloping algebra
• The restricted quantum double of the Yangian
• Shuffle algebras and their integral forms: specialization map approach in types $C_n$ and $D_n$
• The R-matrix formalism for quantized enveloping algebras