Motzkin algebras

We introduce an associative algebra M-k(x) whose dimension is the 2k-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(gl(2)) acting on the k-fold tensor power of the sum of the 1-dimensional and 2-dimensional irreducible U-q(gl(2))-modules. We prove that Mk(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. (C) 2013 Elsevier Ltd. All rights reserved.