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, and it contains the Temperley-Lieb algebra $\TL_k(x)$ as a subalgebra. We prove that for a particular value of $x$, the algebra $\M_k(x)$ is the centralizer algebra of $\uqsl$ acting on the $k$-fold tensor power of the sum of the 1-dimensional and 2-dimensional irreducible $\uqsl$-modules. We show that $\M_k(x)$ is generated by special diagrams $\ell_i, t_i, r_i \ (1 \le i < k)$ and $p_j \ (1 \le j \le k)$, and that it has a factorization into three subalgebras $\M_k(x) = \RP_k \TL_k(x)\, \LP_k$, all of which have dimensions given by Catalan numbers. We define an action of $\M_k(x)$ on Motzkin paths of rank $r$, and in this way, construct a set of indecomposable modules $\C_k^{(r)}$, $0 \le r \le k$. We prove that $\M_k(x)$ is cellular in the sense of Graham and Lehrer and that the $\C_k^{(r)}$ are the left cell representations. We compute the determinant of the Gram matrix of a bilinear form on $\C_k^{(r)}$ for each $r$ and use these determinants to show that $\M_k(x)$ is semisimple exactly when $x$ is not the root of certain Chebyshev polynomials.