Extensions of a commuting pair of quantum toroidal $\mathfrak{gl}_1$
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913097959079936 |
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| author | Feigin, B. Jimbo, M. Mukhin, E. |
| author_facet | Feigin, B. Jimbo, M. Mukhin, E. |
| contents | 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 Drinfeld coproduct. We give examples of $\mathcal{A}_{M,N}$ modules constructed on certain direct sums of tensor products of Fock modules of $\mathcal{E}_1\otimes\check{\mathcal{E}}_1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21750 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extensions of a commuting pair of quantum toroidal $\mathfrak{gl}_1$ Feigin, B. Jimbo, M. Mukhin, E. Quantum Algebra Mathematical Physics 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 Drinfeld coproduct. We give examples of $\mathcal{A}_{M,N}$ modules constructed on certain direct sums of tensor products of Fock modules of $\mathcal{E}_1\otimes\check{\mathcal{E}}_1$. |
| title | Extensions of a commuting pair of quantum toroidal $\mathfrak{gl}_1$ |
| topic | Quantum Algebra Mathematical Physics |
| url | https://arxiv.org/abs/2512.21750 |