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

TL;DR

This paper establishes an algebraic criterion for the density property of affine $SL_2$-manifolds, enabling the construction of many vector fields on their spectra, with applications to Calogero--Moser spaces and quiver varieties.

Contribution

It introduces a new criterion for compatibility of fundamental pairs of derivations, facilitating the proof of algebraic density for key $SL_2$-varieties.

Findings

Proves the algebraic density property for classical Calogero--Moser spaces.

Extends the property to Calogero--Moser spaces with inner degrees of freedom.

Demonstrates the property for smooth cyclic quiver varieties.

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.