Affinization of shifted quantum affine $\mathfrak{gl}_2$

TL;DR

The paper develops a Drinfeld-type, fused-current realization of the quantum toroidal algebra for ${\mathfrak{gl}}_2$ and introduces a shifted affinization ${\mathcal A}_N$ that encodes dominant and non-dominant weight shifts. It constructs ${\mathcal A}_0$ as an extension of two quantum toroidal ${\mathfrak{gl}}_1$-factors, establishes isomorphisms with an extended ${\mathcal E}_2[K]$, and proves a tau- and iota- framework linking ${\mathcal A}_0$ to ${\mathcal E}_2$-modules. The shift ${\mathcal A}_N$ is shown to admit large families of admissible representations, built by tensoring with Fock modules and using coproducts to raise the shift, with a notable connection to extensions of deformed ${W}$-algebras of type ${\mathfrak{gl}}(N+2|1)$ for even $N$. The results lay groundwork for generalizations to ${\mathfrak{gl}}_n$, offering a robust algebraic structure for studying fused currents and shifted toroidal algebras in representation theory and mathematical physics.

Abstract

We give a realization $\mathcal{A}_0$ of quantum toroidal algebra associated to $\mathfrak{gl}_2$ which can be viewed as an affinization of the Drinfeld new realization of quantum affine $\mathfrak{gl}_2$. We use this realization to define an affinization $\mathcal{A}_N$, $N\in{\mathbb Z}$, of shifted quantum affine $\mathfrak{gl}_2$. We construct a large family of representations of dominantly shifted algebra $\mathcal A_N$, $N>0$. The examples of representations with even positive $N$ appear in the study of extensions of deformed $W$-algebras of type $\mathfrak{gl}(N+2|1)$.

Figures (3)

• Figure 1: Fused currents in ${\mathcal{E}}_2$.
• Figure 2: Commutation with $E_1$.
• Figure 3: Commutation with $E_0$.

Theorems & Definitions (44)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6