A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909502244126720 |
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| author | Andrist, Rafael B. Draisma, Jan Freudenburg, Gene Huang, Gaofeng Kutzschebauch, Frank |
| author_facet | Andrist, Rafael B. Draisma, Jan Freudenburg, Gene Huang, Gaofeng Kutzschebauch, Frank |
| contents | Let $B$ be an affine $k$-domain which admits a nontrivial fundamental pair $(D,U)$ of locally nilpotent derivations, i.e., if $E=[D,U]$ then $(D,U,E)$ is an $\mathfrak{sl}_2$-triple. We prove an algebraic criterion, characterizing under which conditions the fundamental pair $(D,U)$ resp. the triple $(D,U,E)$ is compatible in a technical sense that allows us to construct many vector fields on the spectrum of $B$ from the complete ones. This criterion enables us to prove the algebraic density property for the following widely studied classes of $\mathrm{SL}_2$-varieties arising in physics: Classical Calogero--Moser spaces, Calogero--Moser spaces with "inner degrees of freedom'' and smooth cyclic quiver varieties. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_13903 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds Andrist, Rafael B. Draisma, Jan Freudenburg, Gene Huang, Gaofeng Kutzschebauch, Frank Commutative Algebra Algebraic Geometry Complex Variables 13N15, 14J60, 14R20 Let $B$ be an affine $k$-domain which admits a nontrivial fundamental pair $(D,U)$ of locally nilpotent derivations, i.e., if $E=[D,U]$ then $(D,U,E)$ is an $\mathfrak{sl}_2$-triple. We prove an algebraic criterion, characterizing under which conditions the fundamental pair $(D,U)$ resp. the triple $(D,U,E)$ is compatible in a technical sense that allows us to construct many vector fields on the spectrum of $B$ from the complete ones. This criterion enables us to prove the algebraic density property for the following widely studied classes of $\mathrm{SL}_2$-varieties arising in physics: Classical Calogero--Moser spaces, Calogero--Moser spaces with "inner degrees of freedom'' and smooth cyclic quiver varieties. |
| title | A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds |
| topic | Commutative Algebra Algebraic Geometry Complex Variables 13N15, 14J60, 14R20 |
| url | https://arxiv.org/abs/2502.13903 |