The Outer-Temperley-Lieb algebra structure and representation theory

We define a new algebra the Outer-Temperley-Lieb algebra, OTLn(δ), as a fixed ring of the well known Temperley-Lieb algebra, with respect to an automorphism σ reflecting the known diagrammatic representations of the Temperley-Lieb elements in the vertical plane. We define the cell modules of the Outer-Temperley-Lieb algebra and determine that the algebra’s semi-simplicity is dependant entirely on that of the Temperley-Lieb algebra. We are therefore able to give the complete representation theory of the Outer-Temperley- Lieb algebra when it is semi-simple. The induction and restriction of the standard modules to higher and lower rank OTLn(δ) algebras is studied. We also begin a study of the representation theory of OTLn(δ) when it is not semi-simple by describing a large family of homomorphisms between standard modules and conclude with a conjecture on the labelling set of the blocks of the Outer-Temperley-Lieb algebra in the non semi-simple cases.