  Recently, new classes of infinite-dimensional algebras, quiver Yangian (QY)
and shifted QY, were introduced, and they act on BPS states for non-compact
toric Calabi-Yau threefolds. In particular, shifted QY acts on general
subcrystals of the original BPS crystal. A trigonometric deformation called
quiver quantum toroidal algebra (QQTA) was also proposed and shown to act on
the same BPS crystal. Unlike QY, QQTA has a formal Hopf superalgebra structure
which is useful in deriving representations.
  In this paper, we define the shifted QQTA and study a class of their
representations. We define 1d and 2d subcrystals of the original 3d crystal by
removing a few arrows from the original quiver diagram and show how the shifted
QQTA acts on them. We construct the 2d crystal representations from the 1d
crystal representations by utilizing a generalized coproduct acting on
different shifted QQTAs. We provide a detailed derivation of subcrystal
representations of $\mathbb{C}^{3}$, $\mathbb{C}^{3}/\mathbb{Z}_{n}(n\geq 2)$,
conifold, suspended pinch point, and
$\mathbb{C}^{3}/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})$.
