# Solutions of the Yang–Baxter Equation Arising from Brauer Configuration Algebras

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivations

#### 1.2. Contributions

## 2. Background and Related Work

#### 2.1. Yang–Baxter Equation and Its Solutions

- $(G,+)$ is an abelian group.
- $(G,\xb7)$ is a group and$$(a+b)c+c=ac+bc,\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{2.84544pt}{0ex}}a,b,c\in G.$$

**Lemma**

**1**

- 1.
- $a{\lambda}_{a}^{-1}\left(b\right)=b{\lambda}_{b}^{-1}\left(a\right)$.
- 2.
- ${\lambda}_{a}{\lambda}_{{\lambda}_{a}^{-1}\left(b\right)}={\lambda}_{b}{\lambda}_{{\lambda}_{b}^{-1}\left(a\right)}$.
- 3.
- The map $r:G\times G\to G\times G$ defined by $r(x,y)=({\lambda}_{x}\left(y\right),{\lambda}_{{\lambda}_{x}\left(y\right)}^{-1}\left(x\right))$ is a non-degenerate involutive set-theoretical solution of the YBE.

#### 2.2. Multisets and Brauer Configuration Algebras

**Remark**

**1.**

- ${M}_{0}\cap {N}_{0}=\u2300$;
- $${\nu}^{M+N}\left(x\right)=\left\{\begin{array}{cc}{\nu}^{M}\left(x\right),\hfill & \phantom{\rule{5.69046pt}{0ex}}\mathrm{if}\phantom{\rule{5.69046pt}{0ex}}x\in M,\hfill \\ {\nu}^{N}\left(x\right),\hfill & \phantom{\rule{5.69046pt}{0ex}}\mathrm{if}\phantom{\rule{5.69046pt}{0ex}}x\in N;\hfill \end{array}\right.$$
- ${\mathcal{O}}^{M+N}={\mathcal{O}}^{M}+{\mathcal{O}}^{N}$.

#### Brauer Configuration Algebras

- ${Q}_{0}$ is in bijective correspondence with the set of polygons ${\mathcal{M}}_{1}=\{{M}_{1},{M}_{2},\dots ,{M}_{r}\}$, i.e., each vertex ${v}_{i}\in {Q}_{0}$ corresponds to a unique polygon ${M}_{i}\in {\mathcal{M}}_{1}$.
- Each covering ${M}_{i}<{M}_{i+1}$ defined by the orientation $\mathcal{O}$ defines an arrow $\alpha :{v}_{i}\to {v}_{i+1}$, i.e., $s\left(\alpha \right)={v}_{i}$ and $t\left(\alpha \right)={v}_{i+1}$. The cycles ${C}_{\left({x}_{(i,j)}\right)}$ given by a vertex ${x}_{(i,j)}\in {\mathcal{M}}_{0}$ are said to be special cycles.
- Q is bounded by an admissible ideal ${I}_{M}$ (or simply I if no confusion arises) generated by the following three types of relations:
- ${C}_{{x}_{(i,j)}}^{\mu \left({x}_{i,j}\right)}-{C}_{{x}_{i,k}}^{\mu \left({x}_{(i,k)}\right)}$, for any pair of special cycles ${C}_{{x}_{(i,j)}}$ and ${C}_{{x}_{(i,k)}}$ associated with vertices ${x}_{(i,j)},{x}_{(i,k)}\in {M}_{i}$, $1\le i\le r$, fixed (i.e., special cycles defined by vertices in the same polygon are equivalent).
- ${C}_{{x}_{(i,j)}}^{\mu \left({x}_{i,j}\right)}f$, where f is the first arrow of the special cycle ${C}_{{x}_{(i,j)}}$ associated with the vertex ${x}_{(i,j)}$. In particular, if ${l}_{\left({x}_{(i,l)}\right)}$ is a loop associated with a vertex ${x}_{(i,l)}$ with $val\left({x}_{(i,j)}\right)=1$, then a relation of the form ${l}^{\mu \left({x}_{(i,j)}\right)+1}$ also generates the ideal I.
- Quadratic monomial relations of the form $\alpha \beta \in I$, if $\alpha \beta \in \mathbb{F}Q$, $\alpha $ is an arrow contained in an special cycle ${C}_{{x}_{(i,j)}}$, and $\beta $ is contained in an special cycle ${C}_{{x}_{(r,s)}}$ with $i\ne r$.

**Theorem**

**1**

**.**Let $\Lambda =\mathbb{F}Q/I$ be a Brauer configuration algebra induced by a Brauer configuration $M=({\mathcal{M}}_{0},{\mathcal{M}}_{1},\mu ,\mathcal{O})$:

- 1.
- There is a bijection between the set of indecomposable projective modules over Λ and ${\mathcal{M}}_{1}$.
- 2.
- If ${P}_{V}$ is an indecomposable projective module over a BCA Λ defined by a polygon V in ${\mathcal{M}}_{1}$, then $\mathrm{rad}\phantom{\rule{2.84544pt}{0ex}}{P}_{V}=\underset{i=1}{\sum ^{r}}{U}_{i}$, where ${U}_{i}\cap {U}_{j}$ is a simple Λ-module for any $1\le i,j\le r$ and r is the number of (non-truncated) vertices of V.
- 3.
- I is admissible, whereas Λ is a multiserial symmetric algebra. Moreover, if M is connected, then Λ is indecomposable as an algebra.
- 4.
- If $\mathrm{rad}\phantom{\rule{2.84544pt}{0ex}}P$ ($\mathrm{soc}\phantom{\rule{2.84544pt}{0ex}}P$) denotes the radical (socle) of an indecomposable projective module P and ${\mathrm{rad}}^{2}\phantom{\rule{2.84544pt}{0ex}}P\ne 0$, then the number of summands in the heart $\mathrm{rad}\phantom{\rule{2.84544pt}{0ex}}P/\mathrm{soc}\phantom{\rule{2.84544pt}{0ex}}P$ of P equals the number of non-truncated vertices of the polygons in M corresponding to P counting repetitions.
- 5.
- If ${\Lambda}_{M}$ and ${\Lambda}_{{M}^{\prime}}$ are BCAs, induced by Brauer configurations $M=({\mathcal{M}}_{0},{\mathcal{M}}_{1},\nu ,\mathcal{O})$ and ${M}^{\prime}=({\mathcal{M}}_{0}\backslash \left\{h\right\},{\mathcal{M}}_{1}\backslash V\cup {V}^{\prime},\nu ,\mathcal{O})$, where ${V}^{\prime}=V\backslash \left\{h\right\}$, $\left|V\right|\ge 3$, and $val\left(h\right)\mu \left(h\right)=1$, then ${\Lambda}_{M}$ is isomorphic to ${\Lambda}_{{M}^{\prime}}$.

**Proposition**

**1**

**.**Let ${\Lambda}_{M}$ be the Brauer configuration algebra associated with a connected Brauer configuration M. The algebra ${\Lambda}_{M}$ has a length grading induced from the path algebra $\mathbb{F}Q$ if and only if there is an $N\in {\mathbb{Z}}_{>0}$ such that, for each non-truncated vertex, $\delta \in {M}_{0}$, $val\left(\delta \right)\mu \left(\delta \right)=N$.

- ${\mathcal{M}}_{0}=\{1,2,3\}$;
- ${\mathcal{M}}_{1}=\{{V}_{1}=\{3,1,1,1\},{V}_{2}=\{2,1\},{V}_{3}=\{1,2\}\}$;
- $\mathfrak{M}\left(M\right)=31112112$;
- $\nu \left(1\right)=(5,1)$, $\nu \left(2\right)=(2,1)$, $\nu \left(3\right)=(1,2)$;
- Successor sequences: ${S}_{1}={V}_{1}^{\left(1\right)}<{V}_{1}^{\left(2\right)}<{V}_{1}^{\left(3\right)}<{V}_{2}<{V}_{3}$, ${S}_{2}={V}_{2}<{V}_{3}$, ${S}_{3}={V}_{1}$;
- $|{\mathcal{M}}_{0}|=3$, $|{\mathcal{M}}_{1}|=3$, $|{\mathcal{C}}_{M}|=1$;
- ${\mathrm{dim}}_{\mathbb{F}}\phantom{\rule{2.84544pt}{0ex}}{\Lambda}_{M}=29$;
- ${\mathrm{dim}}_{\mathbb{F}}\phantom{\rule{2.84544pt}{0ex}}Z\left({\Lambda}_{M}\right)=8$.

- ${l}_{i}^{1}{l}_{j}^{3}$, ${\left({l}_{i}^{1}\right)}^{2}$, ${\left({l}_{i}^{3}\right)}^{2}$, for all possible values of i and j.
- ${l}_{i}^{3}{\alpha}_{1}^{1}$, ${\alpha}_{1}^{1}{\beta}_{1}^{2}$, ${\beta}_{1}^{2}{\alpha}_{3}^{1}$, ${C}_{1}^{1}\alpha $, ${C}_{2}^{1}{\alpha}^{\prime}$, ${C}_{1}^{1}\sim {C}_{2}^{1}\sim {C}_{3}^{2}$, for all possible special cycles associated with Vertices 1, 2, and 3.

## 3. Main Results

#### 3.1. Brauer Configurations of Type ${\mathfrak{F}}^{(i,j,k)}$

- ${\left({l}_{1}^{j}\right)}^{3}$;
- ${\alpha}_{i}^{0}{l}_{i}^{j}$, ${l}_{i}^{j}{\alpha}_{i}^{0}$, for all possible values of i and j;
- ${\alpha}_{1}^{0}{\alpha}_{2}^{0}{\alpha}_{1}^{0}$, ${\alpha}_{2}^{0}{\alpha}_{1}^{0}{\alpha}_{2}^{0}$.

**Theorem**

**2.**

**Proof.**

- ${a}_{(1,j,1,0)}\in {f}_{(1,j,1,0)}\cap {f}_{(1,j,1,1)}$;
- ${a}_{(i,j,k,1)}\in {f}_{(i,j,k,0)}$;
- ${a}_{(i,j,k,2)}\in {f}_{(i,j,k,k)}$;
- $\{{a}_{i,j,k,1},{a}_{i,j,k,2}\}\subset {\displaystyle \bigcap _{h=1}^{k-1}}{f}_{(i,j,k,h)}$.

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Remark**

**2.**

#### 3.2. Specializations

**Remark**

**3.**

**Remark**

**4.**

- They are piecewise linear and orientation-preserving.
- In the pieces where the maps are linear, the slope is a power of 2.
- Points where slopes change their values are said to be breakpoints, which are dyadic, i.e., they belong to the set $B\times B$, where $B=[0,1]\cap \mathbb{Z}\left[\frac{1}{2}\right]$.

**Lemma**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. Concluding Remarks

#### Future Work

- To determine braces of type ${\mathfrak{H}}_{(i,j,k)}$ associated with Thompson’s groups of type T and V.
- To determine braces based on the Cayley graph of Thompson’s group, $F,T$, and V.
- To give applications of the obtained results in graph energy theory, cryptography, and coding theory.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BCA | Brauer configuration algebra |

${\mathrm{dim}}_{\mathbb{F}}\phantom{\rule{2.84544pt}{0ex}}{\Lambda}_{M}$ | Dimension of a Brauer configuration algebra |

${\mathrm{dim}}_{\mathbb{F}}\phantom{\rule{2.84544pt}{0ex}}Z\left({\Lambda}_{M}\right)$ | Dimension of the center of a Brauer configuration algebra |

$\mathbb{F}$ | Field |

${\mathcal{M}}_{0}$ | Set of vertices of a Brauer configuration M |

$\mathfrak{M}\left(P\right)$ | Brauer message of a Brauer configuration P |

${t}_{n}$ | nth triangular number |

$val\left(\alpha \right)$ | Valency of a vertex $\alpha $ |

${\nu}_{\alpha}$ | $val\left(\alpha \right)\mu \left(\alpha \right)$ |

$w({M}_{i},{f}_{i})$ | The word associated with a polygon $({M}_{i},{f}_{i})$ |

YBE | Yang–Baxter equation |

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**Figure 1.**Main results presented in this paper (targets of red arrows) allow establishing a connection between Brauer configuration algebras and YBE theories.

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**MDPI and ACS Style**

Cañadas, A.M.; Ballester-Bolinches, A.; Gaviria, I.D.M.
Solutions of the Yang–Baxter Equation Arising from Brauer Configuration Algebras. *Computation* **2023**, *11*, 2.
https://doi.org/10.3390/computation11010002

**AMA Style**

Cañadas AM, Ballester-Bolinches A, Gaviria IDM.
Solutions of the Yang–Baxter Equation Arising from Brauer Configuration Algebras. *Computation*. 2023; 11(1):2.
https://doi.org/10.3390/computation11010002

**Chicago/Turabian Style**

Cañadas, Agustín Moreno, Adolfo Ballester-Bolinches, and Isaías David Marín Gaviria.
2023. "Solutions of the Yang–Baxter Equation Arising from Brauer Configuration Algebras" *Computation* 11, no. 1: 2.
https://doi.org/10.3390/computation11010002