Hochschild Cohomology Rings of Temperley-Lieb Algebras

The authors first construct an explicit minimal projective bimodule resolution (P, delta) of the Temperley-Lieb algebra A, and then apply it to calculate the Hochschild cohomology groups and the cup product of the Hochschild cohomology ring of A based on a comultiplicative map Delta : P -> P circle times(A) P. As a consequence, the authors determine the multiplicative structure of Hochschild cohomology rings of both Temperley-Lieb algebras and representation-finite q-Schur algebras under the cup product by giving an explicit presentation by generators and relations.