
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 whose dimension is the -th Motzkin number. The algebra has a basis of "Motzkin diagrams," which are analogous to Brauer and Temperley-Lieb diagrams, and it contains the Temperley-Lieb algebra as a subalgebra. We prove that for a particular value of , the algebra is the centralizer algebra of acting on the -fold tensor power of the sum of the 1-dimensional and 2-dimensional irreducible -modules. We show that is generated by special diagrams and , and that it has a factorization into three subalgebras , all of which have dimensions given by Catalan numbers. We define an action of on Motzkin paths of rank , and in this way, construct a set of indecomposable modules…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Tensor decomposition and applications
