Tied--boxed algebras

We introduce two new algebras that we call \emph{tied–boxed Hecke algebra} and \emph{tied–boxed Temperley–Lieb algebra}. The first one is a subalgebra of the algebra of braids and ties introduced by Aicardi and Juyumaya, and the second one is a tied–version of the well known Temperley–Lieb algebra. We study their representation theory and give cellular bases for them. Furthermore, we explore a strong connection between the tied–boxed Temperley–Lieb algebra and the so–called partition Temperley–Lieb algebra given by Juyumaya. Also, we show that both structures inherit diagrammatic interpretations from a new class of monoids that we call \emph{boxed ramified monoids}. Additionally, we give presentations for the singular part of the ramified symmetric monoid and for the boxed ramified monoid associated to the Brauer monoid.