Gr\"obner-Shirshov bases for Temperley-Lieb algebras of types $B$ and $D$

TL;DR

This paper constructs Gr"obner-Shirshov bases and standard monomials for Temperley-Lieb algebras of types B and D, providing algebraic tools for their analysis.

Contribution

It introduces explicit Gr"obner-Shirshov bases for these algebras, advancing algebraic understanding and computational methods.

Findings

Explicit bases for types B and D

Standard monomials identified

Facilitates algebraic computations in these algebras

Abstract

For Temperley-Lieb algebras of types $B$ and $D$, we construct their Gr\"obner-Shirshov bases and the corresponding standard monomials.