The torsion pair (Y,X)
It is still a torsion pair for the cluster tilted algebra C(B):
For every C(B)-module M, there is
the exact sequence
0 → JM → M → M/JM → 0.
Always: JM belongs to X (since J belongs to X).
If JM ≠ 0, then M/JM belongs to Y (even to Y' = Y \ S).
If M is not a B-module, then both JM and M/JM are non-zero.
This means: J glues together indecomposable B-modules,
but never modules from X or modules from Y.
It puts modules from X on top of modules from Y.