Tensor products, $q$-characters and $R$-matrices for quantum toroidal algebras

TL;DR

This work constructs a robust monoidal framework for quantum toroidal algebras by introducing a horizontal–vertical anti-involution $ψ$ that conjugates the Drinfeld topological coproduct to a well-defined topological coproduct $\triangle^{ψ}_{u}$. This enables a well-defined tensor product on the integrable category $\widehat{\mathcal{O}}_{int}$, with compatibility to Drinfeld polynomials, a generic irreducibility phenomenon for tensor products, and a natural $q$-character calculus that matches the multiplicative structure on the Grothendieck ring. The authors also produce $R$-matrices with spectral parameter satisfying the Yang–Baxter equation, yielding a meromorphic braiding and commuting transfer matrices, thereby extending the affine theory to the toroidal setting. Their results hinge on the extended double affine braid group action, the diagonal subalgebra, and the generalized Miki-type symmetries, and open pathways to toroidal Schur–Weyl dualities, geometric interpretations via Nakajima’s framework, and connections to cluster algebras. Overall, the paper lays foundational monoidal and braiding structures for representations of quantum toroidal algebras, bridging algebraic, geometric, and integrable-system perspectives.

Abstract

We introduce a new topological coproduct $Δ^ψ_{u}$ for quantum toroidal algebras $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ in all untwisted types, leading to a well-defined tensor product on the category $\widehat{\mathcal{O}}_{\mathrm{int}}$ of integrable representations. This is defined by twisting the Drinfeld coproduct $Δ_{u}$ with an anti-involution $ψ$ of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ that swaps its horizontal and vertical quantum affine subalgebras. Other applications of $ψ$ include generalising the celebrated Miki automorphism from type $A$, and an action of the universal cover of $SL_{2}(\mathbb{Z})$. Next, we investigate the ensuing tensor representations of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$, and prove quantum toroidal analogues for a series of influential results by Chari-Pressley on the affine level. In particular, there is a compatibility with Drinfeld polynomials, and the product of irreducibles is generically irreducible. We moreover show that the $q$-character of a tensor product is equal to the product of $q$-characters for its factors. Furthermore, we obtain $R$-matrices with spectral parameter which provide solutions to the (trigonometric, quantum) Yang-Baxter equation, and endow $\widehat{\mathcal{O}}_{\mathrm{int}}$ with a meromorphic braiding. These moreover give rise to a commuting family of transfer matrices for each module.

Figures (5)

• Figure 1: An illustration of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ and its quantum affine subalgebras $\mathcal{U}_{h}$ and $\mathcal{U}_{v}$
• Figure 2: An illustration of $\mathcal{\ddot{B}}$ and its extended affine braid subgroups $\mathcal{B}_{h}$ and $\mathcal{B}_{v}$
• Figure 3: Examples of double affine Coxeter diagrams
• Figure 4: Illustrations of $U_{q}(\mathfrak{g}_{\mathrm{tor}})$ displaying the two finite generating sets
• Figure 5: An illustration of the Yang-Baxter equation

Theorems & Definitions (164)

• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7
• Remark 2.1
• Definition 2.2