Affine BPS algebras, W algebras, and the cohomological Hall algebra of ${\mathbb{A}}^2$

We introduce affinizations and deformations of the BPS Lie algebra associated to a tripled quiver with potential, and use them to precisely determine the $T$-equivariant cohomological Hall algebra ${\mathcal{H}}_{{\mathbb{A}}^2}^T$ of compactly supported coherent sheaves on ${\mathbb{A}}^2$, acted on by a torus $T$. In particular we show that this algebra is spherically generated for all $T$.