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

Michael Finkelberg, Alexander Tsymbaliuk|arXiv (Cornell University)|Nov 29, 2018

Algebraic structures and combinatorial models37 references7 citations

TL;DR

This paper constructs an integral form of shifted quantum affine algebras of type $A$, establishing a Poincaré-Birkhoff-Witt-Drinfeld basis and proving closure under coproduct and shift homomorphisms. It proves that the homomorphism from this integral form to the quantized $K$-theoretic Coulomb branch of a quiver gauge theory is surjective, and identifies the Coulomb branch with the extended quantum universal enveloping algebra in one case.

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).

Motivation & Objective

• To define an integral form of shifted quantum affine algebras of type $A$ over $\mathbb{C}[\boldsymbol{v}, \boldsymbol{v}^{-1}]$.
• To construct a Poincaré-Birkhoff-Witt-Drinfeld basis for these integral forms.
• To prove that the integral form is closed under the coproduct and shift homomorphisms.
• To establish the surjectivity of the homomorphism from the integral form to the quantized $K$-theoretic Coulomb branch of a quiver gauge theory.
• To identify a special case where the Coulomb branch coincides with the extended quantum universal enveloping algebra of type $A$.

Proposed method

• Define the integral form using the RTT presentation of quantum affine $\mathfrak{gl}_n$, extending it to shifted algebras.
• Construct a PBWD basis using quantum minors and Drinfeld generators, ensuring integrality over $\mathbb{C}[\boldsymbol{v}, \boldsymbol{v}^{-1}]$.
• Prove closure under coproduct by analyzing the iterated coproduct $\Delta^n$ and its action on PBW generators.
• Use the Drinfeld-Jimbo presentation and shuffle algebra techniques to relate the integral form to known quantum groups.
• Establish the homomorphism $\overline{\Phi}^\lambda_\mu$ from the integral form to the $K$-theoretic Coulomb branch.
• Verify the surjectivity of $\overline{\Phi}^\lambda_\mu$ via explicit realization and comparison with known Coulomb branch structures.

Experimental results

Research questions

• RQ1 Does there exist an integral form of shifted quantum affine algebras of type $A$ that is closed under the coproduct and shift homomorphisms?
• RQ2 Can a PBWD basis be constructed for this integral form over $\mathbb{C}[\boldsymbol{v}, \boldsymbol{v}^{-1}]$?
• RQ3 Is the homomorphism from the integral form to the quantized $K$-theoretic Coulomb branch of a quiver gauge theory surjective?
• RQ4 In which case does the $K$-theoretic Coulomb branch coincide with the extended quantum universal enveloping algebra of type $A$?
• RQ5 Can the rational (homological) analogues of the results be recovered via this integral form construction?

Key findings

• An integral form $\mathfrak{U}^\mu_{\boldsymbol{v}}[\mathsf{z}_1^{\pm1}, \ldots, \mathsf{z}_N^{\pm1}]$ is constructed for shifted quantum affine algebras of type $A$, with a PBWD basis over $\mathbb{C}[\boldsymbol{v}, \boldsymbol{v}^{-1}]$.
• The integral form is closed under the coproduct and shift homomorphisms, as shown via the iterated coproduct $\Delta^n$ and its action on PBW generators.
• The homomorphism $\overline{\Phi}^\lambda_\mu$ from the integral form to the $K$-theoretic Coulomb branch is surjective, as proven in Theorem 4.32.
• In the case $\mu = 0$, $\lambda = n\omega_{n-1}$, the Coulomb branch is identified with the extended quantum universal enveloping algebra of type $A$.
• The rational (homological) analogues of the results are recovered, confirming earlier results from [KMWY, KTWWY] via a new algebraic framework.
• The PBW theorem for the Yangian is reproven using a novel approach based on coproducts and linear independence of top-degree terms in $\Delta^n(R)$.