  We find a new infinite class of infinite-dimensional algebras acting on BPS
states for non-compact toric Calabi-Yau threefolds. In Type IIA superstring
compactification on a toric Calabi-Yau threefold, the D-branes wrapping
holomorphic cycles represent the BPS states, and the fixed points of the moduli
spaces of BPS states are described by statistical configurations of crystal
melting. Our algebras are "bootstrapped" from the molten crystal
configurations, hence they act on the BPS states. We discuss the truncation of
the algebra and its relation with D4-branes. We illustrate our results in many
examples, with and without compact $4$-cycles.
