  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 $TL_{n}$ that
generalizes to give extensions of $TL_{\infty}$ representations. Finally we
define a generalization of the spin chain representation and conjecture a
generalization of Schur-Weyl duality.

Quantum Algebra;Representation Theory