Motzkin algebras

We introduce an associative algebra Mk(x) whose dimension is the 2k2k-th Motzkin number. The algebra Mk(x) has a basis of “Motzkin diagrams”, which are analogous to Brauer and Temperley–Lieb diagrams. We show for a particular value of xx that the algebra Mk(x) is the centralizer algebra of the quantum enveloping algebra Uq(gl2) acting on the kk-fold tensor power of the sum of the 1-dimensional and 2-dimensional irreducible Uq(gl2)-modules. We prove that Mk(x) is cellular in the sense of Graham and Lehrer and construct indecomposable Mk(x)-modules which are the left cell modules. When Mk(x) is a semisimple algebra, these modules provide a complete set of representatives of isomorphism classes of irreducible Mk(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 Mk(x) is semisimple exactly when xx is not the root of certain Chebyshev polynomials.