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Quantum N-toroidal algebras and extended quantized GIM algebras of N-fold affinization

Yun Gao, Naihuan Jing, Limeng Xia, Honglian Zhang

TL;DR

The paper introduces quantum $N$-toridal algebras $U_q(\mathfrak{g}_{N,tor})$ as a uniform Drinfeld-type quantum $N$-affinization, and relates them to extended quantized GIM algebras of $N$-fold affinization via explicit quotient constructions. It defines a simplified presentation $\mathcal{U}_0(\mathfrak{g}_{N,tor})$ and shows it is isomorphic to $U_q(\mathcal{L}(M))/K$, yielding a systematic bridge to GIM algebras and their Serre relations; for $N=2$ this yields type-dependent (quotient) isomorphisms with the known quantum toroidal algebras, while for $N>2$ the quotient $\overline{U}_q(\mathfrak{g}_{N,tor})$ aligns with $\mathcal{U}_0(\mathfrak{g}_{N,tor})/H_2$. The authors construct vertex representations in the simply-laced case and provide an explicit level-one realization on a Fock space, thereby establishing nontrivial representations and potential categorification avenues. A detailed appendix supplies Dynkin diagrams for $N=2,3$, clarifying the GIM structures. Overall, the work extends the landscape of quantum toroidal-type algebras and clarifies their relation to extended GIM algebras, enabling new representation-theoretic and geometric connections.

Abstract

We introduce the notion of quantum $N$-toroidal algebras as natural generalization of the quantum toroidal algebras as well as extended quantized GIM algebras of $N$-fold affinization. We show that the quantum $N$-toroidal algebras are quotients of the extended quantized GIM algebras of $N$-fold affinization, which generalizes a well-known result of Berman and Moody for Lie algebras.

Quantum N-toroidal algebras and extended quantized GIM algebras of N-fold affinization

TL;DR

The paper introduces quantum -toridal algebras as a uniform Drinfeld-type quantum -affinization, and relates them to extended quantized GIM algebras of -fold affinization via explicit quotient constructions. It defines a simplified presentation and shows it is isomorphic to , yielding a systematic bridge to GIM algebras and their Serre relations; for this yields type-dependent (quotient) isomorphisms with the known quantum toroidal algebras, while for the quotient aligns with . The authors construct vertex representations in the simply-laced case and provide an explicit level-one realization on a Fock space, thereby establishing nontrivial representations and potential categorification avenues. A detailed appendix supplies Dynkin diagrams for , clarifying the GIM structures. Overall, the work extends the landscape of quantum toroidal-type algebras and clarifies their relation to extended GIM algebras, enabling new representation-theoretic and geometric connections.

Abstract

We introduce the notion of quantum -toroidal algebras as natural generalization of the quantum toroidal algebras as well as extended quantized GIM algebras of -fold affinization. We show that the quantum -toroidal algebras are quotients of the extended quantized GIM algebras of -fold affinization, which generalizes a well-known result of Berman and Moody for Lie algebras.

Paper Structure

This paper contains 15 sections, 22 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Extended quantized GIM algebras of $N$-fold affinization
  3. Quantum $N$-toroidal algebras $U_{q}$(${\mathfrak g}_{N,tor}$) $(N\geq 2)$
  4. Quantum $q$-bracket
  5. Quantum $N$-toroidal algebra $U_{q}(\mathfrak{g}_{N, tor})$ via generating functions
  6. Simplified generators and the algebra $\mathcal{U}_0({\mathfrak g}_{N,tor})$
  7. $\mathcal{U}_0$(${\mathfrak g}_{N,tor}$) and the extended quantized GIM algebra of $N$-fold affinizations
  8. Proof of Theorem \ref{['maintheo2-1']}
  9. The algebra $\mathcal{U}_0$($\mathfrak{g}_{2,tor}$)
  10. Proof of Theorem \ref{['maintheo2-1']}
  11. Proof of Theorem \ref{['maintheo2-2']}
  12. Vertex representations of quantum $N$-toroidal algebras $U_{q}(\mathfrak{g}_{N, tor})$ for simply-laced type
  13. Appendix
  14. Dynkin diagrams for the case of $N=2$
  15. Dynkin diagrams for the case of $N=3$

Key Result

Proposition 3.7

For $s\in J$, the following map $\tau_s$ defines an automorphism of $\mathcal{U}_0$(${\mathfrak g}_{N,tor})$: where $i=1, 2, \cdots, n, s'\in J$ and $\epsilon=\pm$ or $\pm1$.

Theorems & Definitions (49)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 39 more