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**A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation.**
*(English)*
Zbl 1074.81036

Let \(V\) be a vector space, and \(R\) a linear operator \(R \in \text{End}(V \otimes V)\), satisfying the Yang-Baxter equation \(R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}\) in \(V^{\otimes 3}\). V. Drinfeld proposed to find and to investigate the set-theoretric solutions \(R\) induced by the map \(X^2 \to X^2\) where \(X\) is a basis of \(V\). This problem appeared connected to many mathematical structures such as quantum binomial algebras, semigroups of I-type and Bieberbach groups, colourings of plane curves and bijective 1-cocycles, semisimple minimal triangular Hopf algebras, dynamical systems and geometric cristals. The unitarity condition for \(R\) implies that \(R\) is bijective. A classification of the non-degenerate unitary solutions \(R\) in terms of associated structure group was given by P. Etingof and T. Schedler, and A. Soloviev, [Duke Math. J. 100, No. 2, 169–209 (1999; Zbl 0969.81030)]. If such a solution \(R\) fixes the diagonal of \(X^2\), then \(R\) is called square-free. This concept has its origin in a class of semigroups of I-type introduced by T. Gateva-Ivanova [Trans. Am. Math. Soc. 343, No. 1, 203–219 (1994; Zbl 0807.16026), J. Algebra 185, No. 3, 710–753 (1996; Zbl 0863.16016)] and called binomial semigroups. These naturally lead to square-free solutions of the Yang-Baxter equation.

T. Gateva-Ivanova proved that every skew-polynomial ring with generating set \(X\) and binomial relations is an Artin-Schelder regular domain of global dimension \(|X|\). Moreover, every such ring gives rise to a non-degenerate unitary set-theoretical solution \(R: X^2 \to X^2\) which fixes the diagonal of \(X^2\). Gateva-Ivanova states the conjecture that, conversely, every such solution \(R\) of the Yang-Baxter equation comes from a skew-polynomial ring with binomial relations. An equivalent conjecture of Etingof, schedler and Soloviev (loc. cit.) says that underlying \(X\) is decomposable. This conjecture was verified in some particular cases.

In the paper under review, using the results of Etingof, the author proves the conjecture in full generality. Futhermore, some new results on the non-degeneracy of unitary solutions \(R: X^2 \to X^2\) of the quantum Yang-Baxter equation are obtained. Besides, several different characterization for the non-degeneracy in terms of cycle sets is given. For a cycle set \(X\) the left multiplications generate a permutation group \(G(X)\) which is in reciprocity with its underlying cycle set \(X\), like the relationship between the Lie groups and Lie algebras. In conclusion the author gives an example of an indecomposable infinite square-free cycle set \(X\), which shows that the decomposability conjecture of Etingof is false when \(X\) is infinite.

T. Gateva-Ivanova proved that every skew-polynomial ring with generating set \(X\) and binomial relations is an Artin-Schelder regular domain of global dimension \(|X|\). Moreover, every such ring gives rise to a non-degenerate unitary set-theoretical solution \(R: X^2 \to X^2\) which fixes the diagonal of \(X^2\). Gateva-Ivanova states the conjecture that, conversely, every such solution \(R\) of the Yang-Baxter equation comes from a skew-polynomial ring with binomial relations. An equivalent conjecture of Etingof, schedler and Soloviev (loc. cit.) says that underlying \(X\) is decomposable. This conjecture was verified in some particular cases.

In the paper under review, using the results of Etingof, the author proves the conjecture in full generality. Futhermore, some new results on the non-degeneracy of unitary solutions \(R: X^2 \to X^2\) of the quantum Yang-Baxter equation are obtained. Besides, several different characterization for the non-degeneracy in terms of cycle sets is given. For a cycle set \(X\) the left multiplications generate a permutation group \(G(X)\) which is in reciprocity with its underlying cycle set \(X\), like the relationship between the Lie groups and Lie algebras. In conclusion the author gives an example of an indecomposable infinite square-free cycle set \(X\), which shows that the decomposability conjecture of Etingof is false when \(X\) is infinite.

Reviewer: Valentina Golubeva (Moskva)

### MSC:

16T20 | Ring-theoretic aspects of quantum groups |

16T25 | Yang-Baxter equations |

17B38 | Yang-Baxter equations and Rota-Baxter operators |

81R12 | Groups and algebras in quantum theory and relations with integrable systems |

81R50 | Quantum groups and related algebraic methods applied to problems in quantum theory |

### Keywords:

quantum Yang-Baxter equation; set-theoretical solutions; indecomposable solution; skew-polynomial ring; cycle sets
Full Text:
DOI

### References:

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