## Braces, radical rings, and the quatum Yang-Baxter equation.(English)Zbl 1115.16022

A set $$X$$ with a binary operation $$\cdot$$ is called a cycle set if $$y\mapsto x\cdot y$$ is invertible and $$(x\cdot y)\cdot(x\cdot z)=(y\cdot x)\cdot(y\cdot z)$$ for all $$x,y,z\in X$$. We say $$X$$ is non-degenerate if $$x\mapsto x\cdot x$$ is bijective. Non-degenerate cycle sets are known to be in one-to-one correspondence with set-theoretical solutions of the quantum Yang-Baxter equation which are non-degenerate and unitary. In the case where $$X=A$$ an Abelian group one can consider the free Abelian group on $$A$$. If $$A$$ has a multiplication $$\cdot$$ which makes $$A$$ a cycle set such that $$a\cdot(b+c)=a\cdot b+a\cdot c$$ and $$(a+b)\cdot c=(a\cdot b)\cdot(a\cdot c)$$, then $$A$$ is a linear cycle set. A non-degenerate cycle set can be embedded into a linear cycle set.
Let $$X$$ be a square-free cycle set, that is $$x\cdot x=x$$ for all $$x$$. (Such an $$X$$ is necessarily non-degenerate.) Let $$Y$$ and $$Z$$ be two finite sub-cycle-sets such that $$Y$$ and $$Z$$ operate transitively on each other. The main theorem in this paper is that $$y\cdot z=y'\cdot z$$ for all $$y,y'\in Y$$ and $$z\in Z$$. As a result, if $$X$$ is finite and has a decomposition $$X=Y\sqcup Z$$ such that $$Y$$ and $$Z$$ operate transitively on each other, then $$X$$ is a bi-cycle. As another consequence, if $$X$$ is a nonempty finite square-free cycle set operating transitively on itself then $$X$$ consists of a single point.
The key to the results in this paper is the relationship between linear cycle sets and braces. Here an Abelian group with a right distributive multiplication if and only if the operation $$a\circ b=ab+a+b$$ (in the free Abelian group on $$A$$) gives a group structure on $$A$$. An explicit one-to-one correspondence is given between braces and linear cycle sets.

### MSC:

 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 81R50 Quantum groups and related algebraic methods applied to problems in quantum theory 16W35 Ring-theoretic aspects of quantum groups (MSC2000) 81R12 Groups and algebras in quantum theory and relations with integrable systems
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### References:

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