Motzkin algebras

We introduce an associative algebra M k ( x ) whose dimension is the 2 k -th Motzkin number. The algebra M k ( x ) has a basis of “Motzkin diagrams”, which are analogous to Brauer and Temperley–Lieb diagrams. We show for a particular value of x that the algebra M k ( x ) is the centralizer algebra of the quantum enveloping algebra U q ( g l 2 ) acting on the k -fold tensor power of the sum of the 1-dimensional and 2-dimensional irreducible U q ( g l 2 ) -modules. We prove that M k ( x ) is cellular in the sense of Graham and Lehrer and construct indecomposable M k ( x ) -modules which are the left cell modules. When M k ( x ) is a semisimple algebra, these modules provide a complete set of representatives of isomorphism classes of irreducible M k ( x ) -modules. We compute the determinant of the Gram matrix of a bilinear form on the cell modules and use these determinants to show that M k ( x ) is semisimple exactly when x is not the root of certain Chebyshev polynomials.