Motzkin Algebras and the $A_n$ Tensor Categories of Bimodules

TL;DR

This paper explores the structure of Motzkin algebras, establishing conditions for factor traces, constructing bimodules, and deriving an $A_n$ tensor category, advancing understanding of algebraic and categorical properties related to these structures.

Contribution

It introduces a new framework connecting Motzkin algebras with $A_n$ tensor categories through bimodule constructions and factor trace conditions.

Findings

Factor trace exists iff D in specified set

Constructed irreducible bimodules over factors

Derived an $A_n$ tensor category from bimodules

Abstract

We discuss the structure of the Motzkin algebra $M_k(D)$ by introducing a sequence of idempotents and the basic construction. We show that $\cup_{k\geq 1}M_k(D)$ admits a factor trace if and only if $D\in \{2\cos(\pi/n)+1|n\geq 3\}\cup [3,\infty)$ and higher commutants of these factors depend on $D$. Then a family of irreducible bimodules over the factors are constructed. A tensor category with $A_n$ fusion rule is obtained from these bimodules.