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Extensions of a commuting pair of quantum toroidal $\mathfrak{gl}_1$

B. Feigin, M. Jimbo, E. Mukhin

TL;DR

The paper defines $\mathcal{A}_{M,N}$ as an extension gluing two commuting quantum toroidal $\mathfrak{gl}_1$ algebras $\mathcal{E}_1$ and $\check{\mathcal{E}}_1$ with parameter constraints $M,N$, recovering a shifted $\mathfrak{gl}_2$ structure when $M=\pm1$. It proposes a coproduct $\Delta_{N_1,N_2}$ into a completed tensor product that extends the standard coproduct on the subalgebra, and develops free-field representations, including $\mathbb{F}_{2;2}$ and Wakimoto-like constructions, as well as tensor products with Fock modules and screened intertwiners to realize level-2-type modules $\mathbb{F}_{2,2;2,2}$. The work analyzes automorphisms, gradings, and special cases (notably $(1,0)$ and $(0,0)$) and discusses the challenges and structure of the coproduct, offering concrete examples that corroborate the conjectured coproduct and hint at connections to shifted and deformed $W$-algebras. This framework exposes a rich algebraic-gluing mechanism for quantum toroidal algebras and opens avenues for further quiver and plane-partition-type interpretations.

Abstract

We introduce a family of algebras $\mathcal{A}_{M,N}$, $M,N\in\mathbb{Z}$, as an extension of a pair of commuting quantum toroidal $\mathfrak{gl}_1$ subalgebras $\mathcal{E}_1,\check{\mathcal{E}}_1$, wherein the parameters are tuned in a specific way according to $M,N$. In the case $M=\pm 1$, algebra $\mathcal{A}_{\pm1,N}$ is a shifted quantum toroidal $\mathfrak{gl}_2$ algebra introduced in [FJM2]. Conjecturally there is a coproduct homomorphism $\mathcal{A}_{M,N_1+N_2}\to\mathcal{A}_{M,N_1}\hat\otimes\mathcal{A}_{M,N_2}$ to a completed tensor product, whose restriction to the subalgebras $\mathcal{E}_1,\check{\mathcal{E}}_1$ coincides with the standard coproduct of the latter. We give examples of $\mathcal{A}_{M,N}$ modules constructed on certain direct sums of tensor products of Fock modules of $\mathcal{E}_1\otimes\check{\mathcal{E}}_1$.

Extensions of a commuting pair of quantum toroidal $\mathfrak{gl}_1$

TL;DR

The paper defines as an extension gluing two commuting quantum toroidal algebras and with parameter constraints , recovering a shifted structure when . It proposes a coproduct into a completed tensor product that extends the standard coproduct on the subalgebra, and develops free-field representations, including and Wakimoto-like constructions, as well as tensor products with Fock modules and screened intertwiners to realize level-2-type modules . The work analyzes automorphisms, gradings, and special cases (notably and ) and discusses the challenges and structure of the coproduct, offering concrete examples that corroborate the conjectured coproduct and hint at connections to shifted and deformed -algebras. This framework exposes a rich algebraic-gluing mechanism for quantum toroidal algebras and opens avenues for further quiver and plane-partition-type interpretations.

Abstract

We introduce a family of algebras , , as an extension of a pair of commuting quantum toroidal subalgebras , wherein the parameters are tuned in a specific way according to . In the case , algebra is a shifted quantum toroidal algebra introduced in [FJM2]. Conjecturally there is a coproduct homomorphism to a completed tensor product, whose restriction to the subalgebras coincides with the standard coproduct of the latter. We give examples of modules constructed on certain direct sums of tensor products of Fock modules of .
Paper Structure (21 sections, 24 theorems, 206 equations, 4 figures)

This paper contains 21 sections, 24 theorems, 206 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Quantum toroidal $\mathfrak{gl}_1$ algebra.
  3. Conventions
  4. Vector representations
  5. Fock representations
  6. Intertwiners.
  7. Fused currents
  8. Algebra $\mathcal{A}_{M,N}$.
  9. Definition
  10. Remarks about $\mathcal{A}_{M,N}$
  11. Special cases
  12. Coproduct issues.
  13. Representations.
  14. Representation $\mathbb{F}_{2;2}$.
  15. Tensor product with Fock representations $\mathcal{F}_1(v)$, $\check\mathcal{F}_1(\check v)$
  16. ...and 6 more sections

Key Result

Lemma 2.3

In $\mathcal{F}_1(u)$ the operators $k^\pm_r(z)$ with $r\geq 2$ act by zero. In addition, we have ∎

Figures (4)

  • Figure 1: Coassociativity.
  • Figure 2: The $\mathfrak{gl}(2n|1)$ Dynkin diagram and labeling.
  • Figure 3: The $qq$-character corresponding to the $\mathfrak{gl}(2n|1)$ vector representation.
  • Figure 4: The $qq$-character $\xi_1^{(3)}$.

Theorems & Definitions (43)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • proof
  • Conjecture 3.4
  • ...and 33 more