CoHA of Cyclic Quivers and an Integral Form of Affine Yangians
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912798661935104 |
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| author | Jindal, Shivang |
| author_facet | Jindal, Shivang |
| contents | We calculate the deformed and non-deformed cohomological Hall algebra (CoHA) of the preprojective algebra for the case of cyclic quivers by studying the Kontsevich-Soibelman CoHA and using tools from cohomological Donaldson-Thomas theory. We show that for the cyclic quiver of length $K$, this algebra is the universal enveloping algebra of the positive half of a certain extension of matrix differential operators on $\mathbb{C}^{*}$, while its deformation gives a positive half of an explicit integral form of Guay's Affine Yangian $\ddot{\mathcal{Y}}_{\hbar_1,\hbar_2}(\mathfrak{gl}(K))$. By the main theorem of Botta-Davison (2023) and Schiffmann-Vasserot (2023), we also determine the Maulik-Okounkov Yangian for the case of cyclic quivers. Furthermore, we explain the construction of factorization coproduct, provide evidence for the strong rationality conjecture, calculate the spherical subalgebra of the non-deformed CoHA for any quiver without loops, recover results about the CoHA of compactly supported semistable sheaves on the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}_{K}$ and identify a commutative algebra inside the additive shuffle algebra associated to the cyclic quiver. We end by conjecturally relating the obtained integral form with the algebra defined by Gaiotto-Rapčák-Zhou, in the context of twisted M-theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02618 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | CoHA of Cyclic Quivers and an Integral Form of Affine Yangians Jindal, Shivang Representation Theory High Energy Physics - Theory Algebraic Geometry Quantum Algebra We calculate the deformed and non-deformed cohomological Hall algebra (CoHA) of the preprojective algebra for the case of cyclic quivers by studying the Kontsevich-Soibelman CoHA and using tools from cohomological Donaldson-Thomas theory. We show that for the cyclic quiver of length $K$, this algebra is the universal enveloping algebra of the positive half of a certain extension of matrix differential operators on $\mathbb{C}^{*}$, while its deformation gives a positive half of an explicit integral form of Guay's Affine Yangian $\ddot{\mathcal{Y}}_{\hbar_1,\hbar_2}(\mathfrak{gl}(K))$. By the main theorem of Botta-Davison (2023) and Schiffmann-Vasserot (2023), we also determine the Maulik-Okounkov Yangian for the case of cyclic quivers. Furthermore, we explain the construction of factorization coproduct, provide evidence for the strong rationality conjecture, calculate the spherical subalgebra of the non-deformed CoHA for any quiver without loops, recover results about the CoHA of compactly supported semistable sheaves on the minimal resolution of the Kleinian singularity $\mathbb{C}^2/\mathbb{Z}_{K}$ and identify a commutative algebra inside the additive shuffle algebra associated to the cyclic quiver. We end by conjecturally relating the obtained integral form with the algebra defined by Gaiotto-Rapčák-Zhou, in the context of twisted M-theory. |
| title | CoHA of Cyclic Quivers and an Integral Form of Affine Yangians |
| topic | Representation Theory High Energy Physics - Theory Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2408.02618 |