The Grothendieck group of a tower of the Temperley–Lieb algebras

Abstract

We provide a diagrammatic approach to compute the algebraic structures on the Grothendieck group of a tower of the Temperley–Lieb algebras.

On Grothendieck ring of the sequence of walled Brauer algebras

“…Using this concept, the structure constants of multiplication over the Grothendieck group can be calculated. The present study aimed to generalize Wang's result [14] to walled Brauer algebras.…”

“…This enables one to consider diagram algebras in the tower framework. Wang [14] studied a tower of Temperley-Lieb algebras TL n and introduced the concept of walled modules to study the branching rules for cell modules of algebras TL m ⊗TL n and TL m+n . Using this concept, the structure constants of multiplication over the Grothendieck group can be calculated.…”

Branching rules for cell modules on a tower of the partition algebras

“…Through the tensor product, on the one hand, for any m, n P N, we can consider the restriction and induction functors between module categories of algebras A m b A n and A m`n ; and on the other hand, the Grothendieck group of the category of finitely generated modules over a tower of algebras can be endowed with an algebra structure. In paper [16], we studied a tower of Temperley-Lieb algebras TL n and introduced the concept of walled modules by studying the characteristics of the restricting Temperley-Lieb diagrams. Through this concept, the branching rules for cell modules of algebras TL m b TL n and TL m`n were obtained and the algebra structure of the Grothendieck group for the tower was calculated.…”

“…In this paper, we draw on the approach used to study the tower of Temperley-Lieb algebras in [16] to study the tower of partition algebras. Precisely, In Section 2, we first recall the concept of partition algebras and the inflation approach used by Xi [17] in proving the cellularity of partition algebras.…”

Semi-simplicity of Temperley-Lieb Algebras of type D

“…For example, the quasi-heredity, semi-simplicity, dimensions of simple modules, Jucys-Murphy elements, Grothendieck group, categorification, block theory, Kazhdan-Lusztig basis and Murphy basis and all kinds of other related topics. We refer the reader to [2,6,7,21,25,27,28] for details. However, the results about other type Temperley-Lieb algebras are few, especially on type D. As far as we know, a Temperley-Lieb algebra of type D is cellular [10] and it has a basis consisting of diagrams [12].…”