A Criterion for the Algebraic Density Property of Affine $SL_2$-Manifolds
Rafael B. Andrist, Jan Draisma, Gene Freudenburg, Gaofeng Huang, Frank Kutzschebauch
TL;DR
The paper provides a precise algebraic criterion for when an SL_2–generated fundamental pair (D,U) or the SL_2 triple (D,U,E) on an affine k-domain B is compatible, phrased in terms of the E–eigengrading of A = ker D: (E,D,U) is compatible iff A_2 ≠ 0 and (D,U) is compatible iff A_1 ≠ 0. It then specializes this criterion to smooth complex affine varieties with SL_2 actions, deriving the algebraic density property for a broad class of SL_2–varieties, including Calogero–Moser spaces and various quiver varieties, by exhibiting explicit complete vector fields and suitable witness functions. The work connects structural results on fundamental pairs with concrete geometric objects, providing new density-property results and a framework for producing large automorphism groups via compatible SL_2 triples. Overall, the results advance the Andersén–Lempert program in the algebraic setting by delivering a clear, checkable criterion and a suite of substantial examples.
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 smooth cyclic quiver varieties.