Shifted Quantum Affine Algebras: Integral Forms in Type A

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 [KMWY, KTWWY] via different techniques).

Shuffle algebra realizations of type A super Yangians and quantum affine superalgebras for all Cartan data

“…Proof. The proof is completely analogous to that of [FT,Proposition 3.44] and proceeds by comparing the matrix coefficients…”

“…They can be viewed as Rees algebras (2.75) of Y (sl(V )) with respect to two standard filtrations on it defined via (2.76), Remark 2.74. The PBW Theorems for Y (sl(V )) and Y (sl(V )), Theorems 2.71 and 2.73, follow from [FT,Theorem B.10,Theorem A.21].…”

Coherent IC-sheaves on type 𝐴_{𝑛} affine Grassmannians and dual canonical basis of affine type 𝐴₁

“…The appearance of the dual canonical basis is natural from yet another point of view. According to [FT19],…”

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

“…A particular PBWD basis of the integral form U v (Lsl n ) (closely related to the RTT integral form of U v (Lgl n )) is used in [FT2] to define an integral form of type A shifted quantum affine algebras of [FT1], see Remarks 2.12, 2.24. An important family of elements of that integral form, which is crucially used in [FT2], appear manifestly via their shuffle realizations (3.39).…”

“…We construct the PBWD bases for U < v (Lsl n ), U > v (Lsl n ) as well as prove their independence of any choices in Theorem 2.19. Following Remark 2.24, the integral form U v (Lsl n ) of the entire U v (Lsl n ) introduced in Definition 2.20 is identified with the RTT integral form U rtt v (Lsl n ), which is used in [FT2] to establish Theorem 2.22. The latter implies the triangular decomposition for U rtt v (Lsl n ) of Corollary 2.23.…”