[Paper Review] An algebra involving braids and ties

Francesca Aicardi, Jesús Juyumaya|arXiv (Cornell University)|Sep 12, 2017

Geometric and Algebraic TopologyMathematics4 references18 citations

TL;DR

This paper introduces and studies the algebra ${\mathcal{E}}_n(u)$, a finite-dimensional algebra over ${\mathbb{C}}(u)$ generated by braid-like elements $T_i$ and tie generators $E_i$, which satisfy relations combining braid group and idempotent relations. The key contribution is the complete classification of the representation theory of ${\mathcal{E}}_3(1)$, showing it is semisimple and decomposes as $4M_1(\mathbb{C}) \oplus 2M_2(\mathbb{C}) \oplus 2M_3(\mathbb{C})$, with dimension 30, using diagrammatic interpretations and morphisms to the group algebra of the hyperoctahedral group.

ABSTRACT

In this note we study a family of algebras with one parameter defined by generators and relations. The set of generators contains the generators of the usual braids algebra, and another set of generators which is interpreted as ties between consecutive strings. We also study the representations theory of the algebra when the parameter is specialized to 1.

Motivation & Objective

• To define and study a new algebra ${\mathcal{E}}_n(u)$ that unifies braid group generators and tie generators, extending Hecke algebra structures.
• To provide a reduced, minimal set of defining relations for ${\mathcal{E}}_n(u)$, removing superfluous relations from earlier formulations.
• To analyze the representation theory of ${\mathcal{E}}_n(u)$ when the parameter $u$ is specialized to 1, particularly for small $n$.
• To establish a diagrammatic interpretation of generators using braids and ties, facilitating the study of linear bases and relations.
• To prove the semisimplicity and explicit decomposition of ${\mathcal{E}}_3(1)$ into matrix algebras over $\mathbb{C}$.

Proposed method

• The algebra ${\mathcal{E}}_n(u)$ is defined via generators $T_1, \dots, T_{n-1}$ and $E_1, \dots, E_{n-1}$, with relations combining braid relations, idempotency of $E_i$, and mixed braid-tie relations.
• A diagrammatic model is introduced where $T_i$ represent crossings and $E_i$ represent ties between adjacent strands, enabling visual reasoning about relations.
• An inductive argument is used to show that ${\mathcal{E}}_n(u)$ is finite-dimensional, with a linear basis constructed recursively via $U_i = \{1\} \cup T_iU_{i-1} \cup E_iU_{i-1} \cup T_iE_iU_{i-1}$.
• The representation theory at $u=1$ is studied by constructing morphisms to the group algebra of the hyperoctahedral group, using a map $\psi$ that sends elements to signed permutations with idempotent actions.
• The linear independence of basis elements is proven by applying evaluation maps $\varphi_0$ and analyzing coefficients in the hyperoctahedral group algebra.
• The decomposition of ${\mathcal{E}}_3(1)$ is derived by showing the dimension is 30 and using the structure of the image under $\psi$, leading to the full matrix decomposition.

Experimental results

Research questions

• RQ1 What is the minimal set of defining relations for the algebra ${\mathcal{E}}_n(u)$ that unifies braids and ties?
• RQ2 How does the representation theory of ${\mathcal{E}}_n(u)$ behave when the parameter $u$ is specialized to 1?
• RQ3 Is the algebra ${\mathcal{E}}_3(1)$ semisimple, and if so, what is its complete decomposition into simple components?
• RQ4 Can a diagrammatic interpretation of the generators as braids and ties provide a systematic way to analyze the algebra’s structure?
• RQ5 What is the dimension of the algebra ${\mathcal{E}}_3(1)$, and how does it relate to the representation theory of the hyperoctahedral group?

Key findings

• The algebra ${\mathcal{E}}_n(u)$ is finite-dimensional, with a basis constructed recursively using the sets $U_i$ defined by $T_i$, $E_i$, and $T_iE_i$ actions on previous generators.
• The defining relations of ${\mathcal{E}}_n(u)$ are reduced to a minimal system, removing two superfluous relations present in the original formulation.
• For $n=2$, a basis of ${\mathcal{E}}_2(u)$ consists of four elements: $\{1, T_1, E_1, T_1E_1\}$.
• For $n=3$, the algebra ${\mathcal{E}}_3(u)$ is linearly spanned by 15 elements of the form $L$, $LE_1$, $LE_2$, $LE_1E_2$, $LE_2T_1$, where $L$ ranges over the 5 elements of the symmetric group $\mathcal{S}_3$.
• The specialization ${\mathcal{E}}_3(1)$ is semisimple and decomposes as $4M_1(\mathbb{C}) \oplus 2M_2(\mathbb{C}) \oplus 2M_3(\mathbb{C})$, with total dimension 30.
• The morphism $\psi$ from ${\mathcal{E}}_3(1)$ to the group algebra of the hyperoctahedral group is injective, confirming the linear independence of the constructed basis.