Maulik-Okounkov quantum loop groups and Drinfeld double of preprojective $K$-theoretic Hall algebras
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909979562213376 |
|---|---|
| author | Zhu, Tianqing |
| author_facet | Zhu, Tianqing |
| contents | In this paper we prove the following results: Given the Drinfeld double $\mathcal{A}^{ext}_{Q}$ of the localised preprojective $K$-theoretic Hall algebra $\mathcal{A}^{+}_{Q}$ of quiver type $Q$ with the Cartan elements, there is a $\mathbb{Q}(q,t_e)_{e\in E}$-Hopf algebra isomorphism between $\mathcal{A}^{ext}_{Q}$ and the localised Maulik-Okounkov quantum loop group $U^{MO}_{q}(\hat{\mathfrak{g}}_{Q})$ of quiver type $Q$. Moreover, we prove the isomorphism of $\mathbb{Z}[q^{\pm1},t_{e}^{\pm1}]_{e\in E}$-algebras between the positive/negative half of the integral Maulik-Okounkov quantum loop group $U_{q}^{MO,\pm,\mathbb{Z}}(\hat{\mathfrak{g}}_{Q})$ with the (opposite) algebra of the integral preprojective (nilpotent) $K$-theoretic Hall algebra $\mathcal{A}^{+,\mathbb{Z}}_{Q}$ ($(\mathcal{A}^{+,nilp,\mathbb{Z}}_{Q})^{op}$) of the same quiver type $Q$. As the application, we prove that one can identify the wall subalgebra $U_{q}^{MO,\mathbb{Z}}(\mathfrak{g}_{w})$ as the root subalgebra $\mathcal{B}_{\mathbf{m},w}^{\mathbb{Z}}$ in the slope subalgebra $\mathcal{B}_{\mathbf{m}}^{\mathbb{Z}}$ as the quasitriangular Hopf $\mathbb{Z}[q^{\pm1},t_e^{\pm1}]_{e\in E}$-algebras. Moreover we use the freeness of the wall subalgebra in MO quantum loop groups to prove the freeness of the preprojective $K$-theoretic Hall algebra for arbitrary torus $\mathbb{C}_q^*\subset A\subset T$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02161 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maulik-Okounkov quantum loop groups and Drinfeld double of preprojective $K$-theoretic Hall algebras Zhu, Tianqing Representation Theory Mathematical Physics Algebraic Geometry Quantum Algebra In this paper we prove the following results: Given the Drinfeld double $\mathcal{A}^{ext}_{Q}$ of the localised preprojective $K$-theoretic Hall algebra $\mathcal{A}^{+}_{Q}$ of quiver type $Q$ with the Cartan elements, there is a $\mathbb{Q}(q,t_e)_{e\in E}$-Hopf algebra isomorphism between $\mathcal{A}^{ext}_{Q}$ and the localised Maulik-Okounkov quantum loop group $U^{MO}_{q}(\hat{\mathfrak{g}}_{Q})$ of quiver type $Q$. Moreover, we prove the isomorphism of $\mathbb{Z}[q^{\pm1},t_{e}^{\pm1}]_{e\in E}$-algebras between the positive/negative half of the integral Maulik-Okounkov quantum loop group $U_{q}^{MO,\pm,\mathbb{Z}}(\hat{\mathfrak{g}}_{Q})$ with the (opposite) algebra of the integral preprojective (nilpotent) $K$-theoretic Hall algebra $\mathcal{A}^{+,\mathbb{Z}}_{Q}$ ($(\mathcal{A}^{+,nilp,\mathbb{Z}}_{Q})^{op}$) of the same quiver type $Q$. As the application, we prove that one can identify the wall subalgebra $U_{q}^{MO,\mathbb{Z}}(\mathfrak{g}_{w})$ as the root subalgebra $\mathcal{B}_{\mathbf{m},w}^{\mathbb{Z}}$ in the slope subalgebra $\mathcal{B}_{\mathbf{m}}^{\mathbb{Z}}$ as the quasitriangular Hopf $\mathbb{Z}[q^{\pm1},t_e^{\pm1}]_{e\in E}$-algebras. Moreover we use the freeness of the wall subalgebra in MO quantum loop groups to prove the freeness of the preprojective $K$-theoretic Hall algebra for arbitrary torus $\mathbb{C}_q^*\subset A\subset T$. |
| title | Maulik-Okounkov quantum loop groups and Drinfeld double of preprojective $K$-theoretic Hall algebras |
| topic | Representation Theory Mathematical Physics Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2511.02161 |