Hochschild cohomology of Temperley--Lieb algebras: a Kauffman diagram perspective

Abstract

We describe Hochschild cohomology of Temperley--Lieb algebras by explicit minimal-resolution coordinates and multilinear operations on Kauffman diagrams. For each doubled-line basic algebra, a finite four-term bimodule complex is verified by a signed-shuffle comparison and an augmented contracting homotopy. We compute cocycles and coboundaries and recover the known cup product and Gerstenhaber bracket in explicit representatives. Over $\mathbb C$, at nonzero root-of-unity loop parameters, we construct a tower-compatible derivation and the marked-circle two-cocycle. Their classes are nonzero on every nonsemisimple block; their cup products therefore represent all positive classes over the centre. Relative full twists give a compact expression for the derivation. Worked planar diagrams distinguish the cup product from insertion and strict cochain equalities from equalities of classes. A finite-lattice section interprets the marked-circle operation as loop-weight response, derives its planar Potts normalization, and exhibits a six-vertex realization. It also shows why passing to a deformable spin representation may annihilate the class with different coefficients. No thermodynamic conclusion is claimed, and the even-strand zero-parameter family remains separate.