Motzkin Algebras

TL;DR

This paper introduces Motzkin algebras, a new associative algebra with diagrammatic basis related to Motzkin numbers, and explores its structure, representations, and semisimplicity conditions.

Contribution

It defines Motzkin algebras, establishes their cellular structure, and connects them to quantum group centralizers, expanding the theory of diagram algebras.

Findings

Motzkin algebra dimension equals the 2k-th Motzkin number.

Motzkin algebra is semisimple for certain parameter values.

Constructs indecomposable modules and describes algebra's cellular structure.

Abstract

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.