The representation theory of the Temperley-Lieb algebras
It is known (Temperley, Lieb, 1971) that the single bond transfer matrices \(u_i\) for the ice model and for the Potts model both satisfy the relations defining the so-called Temperley-Lieb algebra: \(u^2_i=\delta u_i\), \(u_iu_{i\pm 1}u_i=u_i\), \(u_iu_j=u_ju_i\) (\(|i-j|>0\)) where \(\delta\) is a parameter that depends on the model. The main objective of the present paper is to determine all solutions of the Temperley-Lieb relations for matrices with complex entries. As a main result, there is given a finite list of such solutions with the property that any solution can be obtained from this list using two operations: direct sums (decomposable solutions) and change of basis (equivalent solutions). The paper is organized as follows: 1. Diagrams (Yetter, 1989). 2. Paranthesis diagrams (Jones, 1983). 3. Temperley-Lieb algebras (Kaufmann, 1987). 4. Representations. 5. Gram matrices (James, Murphy, 1979; Kazhdan, Lusztig, 1979; Garsia, McLarnan, 1988). 6. Quasi-hereditary algebras (Scott, 1987). 7. Kazhdan-Lusztig representations. 8. The block structure (Jones, 1983). 9. Nonsemisimple case. 10. Indecomposables and resolutions.
scientific article; zbMATH DE number 795321