Tensor products, $q$-characters and $R$-matrices for quantum toroidal algebras
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| Format: | Preprint |
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2025
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| author | Laurie, Duncan |
| author_facet | Laurie, Duncan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_08839 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tensor products, $q$-characters and $R$-matrices for quantum toroidal algebras Laurie, Duncan Quantum Algebra Mathematical Physics Rings and Algebras Representation Theory 17B37, 17B67, 20F36, 81R50 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. |
| title | Tensor products, $q$-characters and $R$-matrices for quantum toroidal algebras |
| topic | Quantum Algebra Mathematical Physics Rings and Algebras Representation Theory 17B37, 17B67, 20F36, 81R50 |
| url | https://arxiv.org/abs/2503.08839 |