SL(2) centralizers

The Temperley-Lieb algebra is the centralizer of (quantum) SL(2) acting on the k-fold tensor product of its natural 2-dimensional representation, and the combinatorics of Catalan numbers play a central role in the underlying representation theory. We tweak this correspondence in two ways and generate interesting algebraic combinatorics. First, in joint work with G. Benkart, we change the underlying module to a direct sum of the 1 and the 2-dimensional irreducible SL(2) modules, and we obtain the combinatorics of the Motzkin numbers. In this setting, the centralizer algebra becomes a diagram algebra that we call the "Motzkin algebra." Second, we consider finite subgroups SL(2) and obtain centralizer algebras whose combinatorics are derived from walks on Dynkin diagrams of type ADE via the McKay correspondence.