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 TLn that generalizes to give extensions of TL∞ representations. Finally, we define a generalization of the spin chain representation and conjecture a generalization of Schur-Weyl duality.