Dual presentation and linear basis of the Temperley-Lieb algebras

The braid group $B_n$ maps homomorphically into the Temperley-Lieb algebra ${\text{TL}}_n$. It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group $B_n$ form a basis for the vector space underlying the Temperley-Lieb algebra ${\text{TL}}_n$. In this paper, we establish that there is a dual presentation of Temperley-Lieb algebras that corresponds to the dual presentation of braid groups, and then give a simple geometric proof for Zinno’s theorem, using the interpretation of simple elements as non-crossing partitions.