  The Temperley-Lieb algebra is a fundamental component of SU(2) topological
quantum field theories. We construct chain complexes corresponding to minimal
idempotents in the Temperley-Lieb algebra. Our results apply to the framework
which determines Khovanov homology. Consequences of our work include
semi-orthogonal decompositions of categorifications of Temperley-Lieb algebras
and Postnikov decompositions of all Khovanov tangle invariants.
