Question:
In what sense is the Temperley-Lieb algebra $ TL_n $ related to (a representation of?) the braid group $ B_n $?
For example, is $ TL_n $ the algebra of matrices generated by the image of $ B_n $ under the Burau representation?
Context:
I've heard that the Temperley-Lieb algebra is related to the braid group, but I'm not sure how. $ B_n $ and $ TL_n $ both have $ n-1 $ generators. Some of the relations on the $ TL_n $ generators look similar to the $ B_n $ relations, but they aren't quite identical.
Not sure this is exactly the kind of answer your looking for but hopefully helpful nonetheless.
$TL_n$ has extra relations on top of those satisfied by the braid group. One key difference is the existence of a complex valued parameter, often denoted $\delta$ in the definition of $TL_n$, namely $e_i^2 = \delta e_i$ for all generators $e_1,\dots,e_{n-1}$. Another thing to note is that some presentations of $TL_n$ feature non-invertible generators. One way to explore this relationship is by looking at the natural map $B_n \to TL_n$ which (because the relations defining $B_n$ are included in those defining $TL_n$) is a homomorphism. In fact, one can identify certain "nice" elements of $B_n$ which map to a basis for $TL_n$ under this homomorphism. For more details, check out this paper by Eon-Kyung Lee and Sang Jin Lee.