On the Representation theory of the Infinite Temperley-Lieb algebra

We begin the study of the representation theory of the infinite Temperley-Lieb algebra. We fully classify its finite dimensional representations, then introduce infinite link state representations and classify when they are irreducible or indecomposable. We also define a construction of projective indecomposable representations for $T{L}_n$ that generalizes to give extensions of $T{L}_{\mathrm{\infty }}$ representations. Finally we define a generalization of the spin chain representation and conjecture a generalization of Schur-Weyl duality.