Brauer Analysis of Thompson’s Group F and Its Application to the Solutions of Some Yang–Baxter Equations
Abstract
:1. Introduction
Contributions
2. Preliminaries
2.1. Background
2.2. Thompson’s Group F
- F is torsion-free, contains free semi-group on two generators, and contains no nonabelian free group.
- Every subgroup of F is finite rank free abelian or contains an infinite rank free abelian subgroup.
- Every proper quotient of F is abelian.
- is simple and .
2.3. Tree Operations
- is defined in such a way that if v is a tree root and w is a vertex in another tree, link the trees containing v and w by adding the edge , making w the parent of v.
- divides a given rooted tree into two new trees by deleting the edge from a vertex v (which is not a tree root) to its parent.
- turns a tree containing a vertex v inside out by making v the root of the tree.
2.4. Yang–Baxter Equation and Its Solutions
- is an abelian group;
- is a group.
- 1.
- .
- 2.
- .
- 3.
- The map defined by is a non-degenerate involutive set-theoretical solution of the YBE.
2.5. Multisets and Brauer Configuration Algebras
- Each letter has associated a linear ordered set or successor sequence , where , is the collection of all multisets containing m and , with , and , , if .
- If is the size of the multiset , then , .
- There exists a map ) such that , where is said to be the valency of m. If and (), then (). In particular, the successor sequence associated with m is , if .
- , if and are special cycles at vertices and contained in the same polygon . These are relations of type .
- , where is a special cycle at vertex and f is its first arrow. In particular, if is an arrow defined by with , then is a relation of this type provided that in such a case, is a special cycle at . These relations are said to be of type .
- , where and are arrows in defined by successor sequences and with , and . These relations are said to be of type .
- Any Brauer configuration algebra induced by a Brauer configuration is multiserial, and there exists a bijective correspondence between the set of indecomposable projective -modules and the set of multisets or polygons .
- The number of summands in the heart of an indecomposable projective -module equals the number of nontruncated vertices in the corresponding polygon. Where, () denotes the Jacobson radical (socle) of the indecomposable projective -module P.
- We note that any graph gives rise to a Brauer configuration of type M, where and . Cañadas et al. [25] proved that the covering graph induced by a Brauer configuration defined by a graph is isomorphic to if and only if is a disjoint union of copies of connected hair graphs of type , where is an n-point cycle and .
- , .
- .
- .
- , , .
- , , for the remaining vertices or letters.
- The Brauer configuration algebra induced by the Brauer configuration is indecomposable as an algebra, .
- Figure 4 shows Brauer quivers and . The following are examples of relations contained in the admissible ideal . Relations in are defined in the same fashion.
- 1.
- , , for all possible values of p and q.
- 2.
- , .
- 3.
- .
- 4.
- .
- 5.
- .
- 6.
- .
- 7.
- ; ; .
- For fixed, it holds that .
- and are isomorphic as algebras.
- is the Brauer configuration algebra induced by the generator of Thompson’s group F. Thus, .
- For fixed, it holds that . Note that there are three vertices with valency one and four polygons (see Formula (18)).
2.6. Entropy of a Graph
- .
- .
- The degree of the vertices gives their valency values, i.e., , and , for .
- The multiplicity values of the elements of M (vertices of the Brauer configuration ) are , , , and .
- Circular orderings associated with vertices , areThese circular orderings define the orientation . The symbol in a circular ordering means that occurs q times at the polygon (edge) .
- The Brauer quiver induced by the Brauer configuration is such that the set of vertices and there exists an arrow if and only if there exists a relation in the circular ordering associated with the vertex (see Figure 5).
- The Brauer configuration algebra is the bound quiver algebra induced by the Brauer configuration , the following are examples of relations contained in the admissible ideal .
- 1.
- , , , .
- 2.
- , , , .
- 3.
- if , for all values of r and s.
- The Brauer configuration algebra is indecomposable as an algebra and reduced, provided that the Brauer quiver is connected and , for any .
- Since Γ is a hair graph of type , then the covering graph is isomorphic to Γ, whose entropies , , and (the entropy of the Brauer configuration ) are as follows:
- 1.
- .
- 2.
- .
- 3.
- , provided that . In [25], Cañadas et al. proved that if Γ is a Dynkin or Euclidean diagram, then allows for giving approximations of .
- The dimensions of the Brauer configuration algebra and its corresponding center are given by the following identities:
3. Main Results
3.1. Brauer Analysis of Thompson’s Group F
- 1.
- is indecomposable as an algebra.
- 2.
- .
- 3.
- .
- 4.
- .
- 5.
- .
- 6.
- .
- is indecomposable as an algebra provided that the binary tree , which induces , is connected.
- Note that the number of edges of the binary tree is . Furthermore, , , and . Then, .
- .
- .
- Note that, , then .
- , then . We are done.
3.2. Solutions of the Yang–Baxter Equation Arising from Covering Graphs
- If , , , and , then=.On the other hand,.
- If , , with h, i, and f arbitraries, then=.On the other hand,.
3.3. Brauer Analysis of Thompson’s Group F via Christoffel Words
- 1.
- , .
- 2.
- .
- 3.
- .
- 1.
- is indecomposable as an algebra.
- 2.
- .
- 3.
- .
- 4.
- The covering graph is a 4-point linear path.
- If denotes the polygon defined by the word , , then , hence the Brauer quiver induced by the Brauer configuration is connected. Therefore, the algebra is indecomposable as an algebra.
- We note that, if , and are rational numbers whose Christoffel words are , and , respectively. Then, , , , and . Therefore, , , , . Furthermore, , , , . Where, , denotes the number of occurrences of the number h in the polygon . Thus, and , then .
- Note that, for any , and , and the number of loops in the Brauer quiver induced by the Brauer configuration is . Then, .
- , where is the chain induced by the expansions of .
4. Conclusions
- Applying Brauer analysis to generators of Thompson’s group T and V is an open problem. Such an analysis can include giving values of the super edge-magic total strength of covering graphs, as Deepthi defined in [45].
- Another task for the future consists of giving new solutions of the Yang–Baxter equation arising from the Brauer configuration algebras induced by generators of Thompson’s group T and V.
- Applying Brauer analysis to quantum circuits is a task to be addressed in the future, assuming quantum gates as elements of the ground set M and quantum circuits as elements of a collection of multisets denoted in this paper.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
Binary tree defined by Thompson’s group generator | |
Christoffel path | |
Covering graph of a Brauer quiver | |
Dimension of a Brauer configuration algebra | |
Dimension of the center of a Brauer configuration algebra | |
Entropy of a graph | |
Entropy of a Brauer configuration induced by a graph | |
Hair graph | |
Brauer configuration algebra | |
Rotation graph whose set of vertices are n-vertex binary trees | |
M | Set of vertices of a Brauer configuration |
Collection of polygons of a Brauer configuration | |
Brauer configuration | |
Multiplicity function of a Brauer configuration | |
Orientation of a Brauer configuration | |
valency of a vertex x | |
YBE | Yang–Baxter equation |
Center of a Brauer configuration algebra |
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Cañadas, A.M.; Rodríguez-Nieto, J.G.; Salazar-Díaz, O.P.; Velásquez, R.; Giraldo, H. Brauer Analysis of Thompson’s Group F and Its Application to the Solutions of Some Yang–Baxter Equations. Mathematics 2025, 13, 1127. https://doi.org/10.3390/math13071127
Cañadas AM, Rodríguez-Nieto JG, Salazar-Díaz OP, Velásquez R, Giraldo H. Brauer Analysis of Thompson’s Group F and Its Application to the Solutions of Some Yang–Baxter Equations. Mathematics. 2025; 13(7):1127. https://doi.org/10.3390/math13071127
Chicago/Turabian StyleCañadas, Agustín Moreno, José Gregorio Rodríguez-Nieto, Olga Patricia Salazar-Díaz, Raúl Velásquez, and Hernán Giraldo. 2025. "Brauer Analysis of Thompson’s Group F and Its Application to the Solutions of Some Yang–Baxter Equations" Mathematics 13, no. 7: 1127. https://doi.org/10.3390/math13071127
APA StyleCañadas, A. M., Rodríguez-Nieto, J. G., Salazar-Díaz, O. P., Velásquez, R., & Giraldo, H. (2025). Brauer Analysis of Thompson’s Group F and Its Application to the Solutions of Some Yang–Baxter Equations. Mathematics, 13(7), 1127. https://doi.org/10.3390/math13071127