Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An abstract approach to algebras of braids and ties

Riccardo Fasano, Domenico Fiorenza, Paolo Papi

TL;DR

The paper tackles unifying and extending algebras of braids and ties across Coxeter types by introducing Marin data and the universal Marin algebra $\mathcal{E}^{(W,S)}(A,\rho,a)$, defined as a quotient of $A \rtimes k[B_W]$ by Hecke-type relations. It proves a central freeness criterion, shows how to lift Hecke algebras via Marin-Iwahori-Hecke monoids and Juyumaya maps, and establishes functoriality that preserves algebraic structure under morphisms of Marin data. The framework recovers Marin’s $\mathcal{C}_W$ algebras and, in type $A$, the known braids and ties algebras, while producing new algebras for types $B$, $D$, and affine settings. By providing explicit constructions, rank formulas, and a diagrammatic interpretation, the work offers a cohesive, extensible approach to braids and ties algebras with potential connections to framization, deframization, and broader representation-theoretic contexts $($e.g.$, H(W,S)(u_s)$ and $\mathcal{E}_n(u,v))$.

Abstract

Generalizing work of Marin [12], we construct in a unified way all the "braids and ties'' algebras available in literature and new ones.

An abstract approach to algebras of braids and ties

TL;DR

The paper tackles unifying and extending algebras of braids and ties across Coxeter types by introducing Marin data and the universal Marin algebra , defined as a quotient of by Hecke-type relations. It proves a central freeness criterion, shows how to lift Hecke algebras via Marin-Iwahori-Hecke monoids and Juyumaya maps, and establishes functoriality that preserves algebraic structure under morphisms of Marin data. The framework recovers Marin’s algebras and, in type , the known braids and ties algebras, while producing new algebras for types , , and affine settings. By providing explicit constructions, rank formulas, and a diagrammatic interpretation, the work offers a cohesive, extensible approach to braids and ties algebras with potential connections to framization, deframization, and broader representation-theoretic contexts e.g. and .

Abstract

Generalizing work of Marin [12], we construct in a unified way all the "braids and ties'' algebras available in literature and new ones.

Paper Structure

This paper contains 4 sections, 20 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. Marin Algebras
  3. Examples
  4. Braids and ties algebras

Key Result

Theorem 1.1

Let $(W,S)$ and $(U,T)$ be Coxeter systems and let $(\phi,e)$ be a Juyumaya map between them. Let $\rho_\phi$ denote the $W$-action on $\Sigma(\Phi_U)$ induced by $\phi\colon W\to U$ and by the canonical action of $U$ on $\Phi_U$. Then $(\Sigma(\Phi_U)_{[e]},\rho_\phi,e)$ is a Marin-Iwahori-Hecke mo

Theorems & Definitions (77)

  • Theorem 1.1
  • Definition 2.1
  • Example 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • ...and 67 more