On the representation theory of the infinite Temperley–Lieb algebra

Abstract

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[Formula: see text] that generalizes to give extensions of TL[Formula: see text] representations. Finally, we define a generalization of the spin chain representation and conjecture a generalization of Schur–Weyl duality.