Constructing a free resolution and remarks on bar homology of Temperley-Lieb algebra $T{L}_3$

Abstract

In this paper I try to construct the Anick's resolution for the Temperley-Lieb algebra $T{L}_3$ and bar homology $To{r}_{\ast }^{T{L}_3}(\mathbb{C},\mathbb{C})$ and shows that how $\tau \in \mathbb{C}$ given in the definition of $T{L}_3$ is actively involved in the differentials both in free resolution and bar homology. It suggests that the bar homology depends on the nature of $\tau$.

6 pages, This paper mentions some interesting open problems regarding Temperley-Lieb algebra and its connection with braid group. It would be also interesting to compute some higher homotopy associative structure maps of the algebra