Uncoiled affine Temperley–Lieb algebras and their Wenzl–Jones projectors

Affine and periodic Temperley-Lieb algebras are families of diagrammatic algebras that find diverse applications in mathematics and physics. These algebras are infinite dimensional, yet most of their interesting modules are finite. In this paper, we introduce finite quotients for these algebras, which we term uncoiled affine Temperley-Lieb algebras and uncoiled periodic Temperley-Lieb algebras. We study some of their properties, including their defining relations, their description with diagrams, their dimensions, and their relations with affine and skew sandwich cellular algebras. The uncoiled algebras all have finitely many one-dimensional modules. We construct a family of Wenzl-Jones idempotents, each of which projects onto one of these one-dimensional modules. Our construction is explicit and uses the similar projectors for the ordinary Temperley--Lieb algebras, as well as the diagrammatic description of the uncoiled algebras in terms of sandwich diagrams. We also discuss the Markov traces for the uncoiled algebras and their evaluations on the newly defined projectors, and find expressions involving Chebyshev polynomials of the first kind.