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# a Skew Brace?

Communicated by *Notices* Associate Editor Han-Bom Moon

A skew brace is a fascinating mathematical structure that involves a pair of compatible groups sharing the same underlying set. The notion of skew braces was inspired by a class of rings introduced by Jacobson in 1945, which we will discuss in more detail later. As algebraic structures, skew braces exhibit similarities to both groups and rings.

What makes skew braces particularly interesting is their ability to serve as an algebraic framework for exploring combinatorial solutions to the Yang–Baxter equation. By delving into the world of skew braces, we can uncover new insights and approaches to understanding this fundamental equation.

Let us begin by understanding what we mean by combinatorial solutions to the Yang–Baxter equation.

We are interested in pairs where , is just a set and is a bijective map that satisfies a specific equation in called the Yang–Baxter equation: ,

This equation may look a bit abstract, but there is a nice way to think about it:

The arrangements of strings, representing the Yang–Baxter equation, should be read from top to bottom, with the crossing symbolizing the application of the map and the straight line representing the identity mapping. The picture itself is self-explanatory.

Because of this braiding-like behavior, the equation is also known as the *braid equation*.

The identity map on satisfies the Yang–Baxter equation. However, without additional assumptions, the task of finding solutions becomes highly unpredictable. Therefore, we will focus on solutions that meet specific extra assumptions. Given the combinatorial nature of the problem and our intention to utilize group theory, it is compelling to investigate the following intriguing class of solutions. We say that a solution where ,

is *nondegenerate* if the maps and are bijective, for all .

With the nondegeneracy assumption, exploring solutions to the Yang–Baxter equation becomes even more intriguing as we can now leverage groups that inherently act on our solutions. This opens up new avenues for investigation. By incorporating group actions, we gain a richer understanding of the equation’s behavior and its connections to various mathematical structures.

We can find many examples of solutions:

- (a)
If and are commuting bijections, then is a solution. In particular, the flip map is a solution.

- (b)
If is a group, then is a solution.

In one of the groups, we embrace the use of additive notation, even when our group is not necessarily assumed to be abelian.

Radical rings have their roots in Jacobson’s work. Subsequently, Rump unearthed an algebraic structure encompassing Jacobson radical rings as examples and coined the term *braces*. This structure proposed by Rump was further expanded upon in 3, leading to what we currently know as *skew braces*.

Many familiar mathematical objects have skew brace structures. For example, groups trivially produce skew braces.

Now let us explore Jacobson radical rings. Imagine we have a ring In this ring, we can define a new operation—called the .*Jacobson circle operation*—that takes two elements, let us say and and maps them to ,

What is surprising is that this operation is always associative with the zero of the ring being its neutral element. When is a group, we say that is a *radical ring*. For instance, nilpotent rings, such as rings of strictly upper triangular matrices, are Jacobson radical rings.

Now, here is the exciting part discovered by Rump: Jacobson radical rings are also examples of skew braces. When considering a radical ring, the combination of its addition with the Jacobson circle operation transforms the ring into a skew brace.

It is time to unveil the fascinating connection between skew braces and solutions to the Yang–Baxter equation.

The reader is encouraged to explore the solutions obtained by applying Theorem 4 to the skew braces described in Examples 2 and 3.

For a solution we define the ,*structure group* of as the group with generators and relations

whenever .

In the upcoming example, we will express permutations as products of disjoint cycles. For instance, the symbol denotes the bijective mapping from to where , is mapped to , is mapped to and , is mapped to .

The linear representation we observed in the previous example was first found by Etingof, Schedler, and Soloviev and can be now explained by the theory of skew braces.

The previous theorem reveals a profound connection between solutions and skew braces. It uncovers a hidden bridge that connects the world of combinatorial properties of solutions to the algebraic properties of skew braces and vice versa.

It is remarkable how seemingly different realms of mathematics are intertwined, providing us with new opportunities to explore the intricate relationship between combinatorial and algebraic concepts.

Skew braces offer an appealing framework to explore various mathematical problems that may not initially seem connected to the solutions of the Yang–Baxter equation. Examples are the links between skew braces and Lie theory or Hopf–Galois structures. It is amazing how these apparently unrelated concepts can come together and inspire new insights in mathematics.

## References

- [1]
- Ferran Cedó, Eric Jespers, and Jan Okniński,
*Braces and the Yang-Baxter equation*, Comm. Math. Phys.**327**(2014), no. 1, 101–116, DOI 10.1007/s00220-014-1935-y. MR3177933,## Show rawAMSref

`\bib{MR3177933}{article}{ author={Ced\'{o}, Ferran}, author={Jespers, Eric}, author={Okni\'{n}ski, Jan}, title={Braces and the Yang-Baxter equation}, journal={Comm. Math. Phys.}, volume={327}, date={2014}, number={1}, pages={101--116}, issn={0010-3616}, review={\MR {3177933}}, doi={10.1007/s00220-014-1935-y}, }`

- [2]
- Pavel Etingof, Travis Schedler, and Alexandre Soloviev,
*Set-theoretical solutions to the quantum Yang-Baxter equation*, Duke Math. J.**100**(1999), no. 2, 169–209, DOI 10.1215/S0012-7094-99-10007-X. MR1722951,## Show rawAMSref

`\bib{MR1722951}{article}{ author={Etingof, Pavel}, author={Schedler, Travis}, author={Soloviev, Alexandre}, title={Set-theoretical solutions to the quantum Yang-Baxter equation}, journal={Duke Math. J.}, volume={100}, date={1999}, number={2}, pages={169--209}, issn={0012-7094}, review={\MR {1722951}}, doi={10.1215/S0012-7094-99-10007-X}, }`

- [3]
- L. Guarnieri and L. Vendramin,
*Skew braces and the Yang-Baxter equation*, Math. Comp.**86**(2017), no. 307, 2519–2534, DOI 10.1090/mcom/3161. MR3647970,## Show rawAMSref

`\bib{MR3647970}{article}{ author={Guarnieri, L.}, author={Vendramin, L.}, title={Skew braces and the Yang-Baxter equation}, journal={Math. Comp.}, volume={86}, date={2017}, number={307}, pages={2519--2534}, issn={0025-5718}, review={\MR {3647970}}, doi={10.1090/mcom/3161}, }`

- [4]
- Jiang-Hua Lu, Min Yan, and Yong-Chang Zhu,
*On the set-theoretical Yang-Baxter equation*, Duke Math. J.**104**(2000), no. 1, 1–18, DOI 10.1215/S0012-7094-00-10411-5. MR1769723,## Show rawAMSref

`\bib{MR1769723}{article}{ author={Lu, Jiang-Hua}, author={Yan, Min}, author={Zhu, Yong-Chang}, title={On the set-theoretical Yang-Baxter equation}, journal={Duke Math. J.}, volume={104}, date={2000}, number={1}, pages={1--18}, issn={0012-7094}, review={\MR {1769723}}, doi={10.1215/S0012-7094-00-10411-5}, }`

- [5]
- Wolfgang Rump,
*Braces, radical rings, and the quantum Yang-Baxter equation*, J. Algebra**307**(2007), no. 1, 153–170, DOI 10.1016/j.jalgebra.2006.03.040. MR2278047,## Show rawAMSref

`\bib{MR2278047}{article}{ author={Rump, Wolfgang}, title={Braces, radical rings, and the quantum Yang-Baxter equation}, journal={J. Algebra}, volume={307}, date={2007}, number={1}, pages={153--170}, issn={0021-8693}, review={\MR {2278047}}, doi={10.1016/j.jalgebra.2006.03.040}, }`

- [6]
- Alexander Soloviev,
*Non-unitary set-theoretical solutions to the quantum Yang-Baxter equation*, Math. Res. Lett.**7**(2000), no. 5-6, 577–596, DOI 10.4310/MRL.2000.v7.n5.a4. MR1809284,## Show rawAMSref

`\bib{MR1809284}{article}{ author={Soloviev, Alexander}, title={Non-unitary set-theoretical solutions to the quantum Yang-Baxter equation}, journal={Math. Res. Lett.}, volume={7}, date={2000}, number={5-6}, pages={577--596}, issn={1073-2780}, review={\MR {1809284}}, doi={10.4310/MRL.2000.v7.n5.a4}, }`

## Credits

Figure 1 and photo of Leandro Vendramin are courtesy of Leandro Vendramin.