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arXiv · 2502.13903

A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds

Rafael B. Andrist·Jan Draisma·Gene Freudenburg·Gaofeng Huang·Frank Kutzschebauch
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Abstract

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 a smooth cyclic quiver variety.

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BibTeXRIS

Rafael B. Andrist, Jan Draisma, Gene Freudenburg, Gaofeng Huang, Frank Kutzschebauch. 2025-02-19. A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds. https://arxiv.org/abs/2502.13903

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