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Let $(W,S)$ be an affine Coxeter system of type $\widetilde{B}$ or$\widetilde{D}$ and ${\rm TL}(W)$ the corresponding generalized Temperley-Liebalgebra. In this paper we define an infinite dimensional associative algebramade of decorated diagrams that is isomorphic to ${\rm TL}(W)$. Moreover, wedescribe an explicit basis for such an algebra consisting of special decorateddiagrams that we call admissible. Such basis is in bijective correspondencewith the classical monomial basis of the generalized Temperley-Lieb algebraindexed by the fully commutative elements of $W$.

Diagrammatic representations of Generalized Temperley-Lieb algebras of affine type $\widetilde{B}$ and $\widetilde{D}$