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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.

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
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[1] V.G. Drinfeld, On some unsolved problems in quantum group theory, in: P.P. Kulish (Ed.), Quantum Groups (Leningrad, 1990), Lecture Notes in Mathematics, Vol. 1510, Springer, Berlin, 1992, pp. 1-8.; V.G. Drinfeld, On some unsolved problems in quantum group theory, in: P.P. Kulish (Ed.), Quantum Groups (Leningrad, 1990), Lecture Notes in Mathematics, Vol. 1510, Springer, Berlin, 1992, pp. 1-8. · Zbl 0765.17014
[2] P. Etingof, Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, math. QA/0112278 (2001).; P. Etingof, Geometric crystals and set-theoretical solutions to the quantum Yang-Baxter equation, math. QA/0112278 (2001).
[3] Etingof, P.; Gelaki, S., A method of construction of finite-dimensional triangular semisimple Hopf algebras, Math. Res. Lett., 5, 551-561 (1998) · Zbl 0935.16029
[4] Etingof, P.; Schedler, T.; Soloviev, A., Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Math. J., 100, 169-209 (1999) · Zbl 0969.81030
[5] Gateva-Ivanova, T., Noetherian properties of skew-polynomial rings with binomial relations, Trans. Amer. Math. Soc., 343, 203-219 (1994) · Zbl 0807.16026
[6] Gateva-Ivanova, T., Skew polynomial rings with binomial relations, J. Algebra, 185, 710-753 (1996) · Zbl 0863.16016
[7] T. Gateva-Ivanova, A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation, Preprint.; T. Gateva-Ivanova, A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation, Preprint. · Zbl 1065.16037
[8] T. Gateva-Ivanova, Regularity of skew-polynomial rings with binomial relations, Talk at the International Algebra Conference, Miskolc, Hungary, 1996.; T. Gateva-Ivanova, Regularity of skew-polynomial rings with binomial relations, Talk at the International Algebra Conference, Miskolc, Hungary, 1996. · Zbl 0863.16016
[9] Gateva-Ivanova, T.; Van den Bergh, M., Semigroups of I-type, J. Algebra, 206, 97-112 (1998) · Zbl 0944.20049
[10] Jespers, E.; Okniński, J., Binomial semigroups, J. Algebra, 202, 250-275 (1998) · Zbl 0910.20038
[11] G. Laffaille, Quantum binomial algebras, preprint.; G. Laffaille, Quantum binomial algebras, preprint. · Zbl 1011.16029
[12] Lu, J.-H.; Yan, M.; Zhu, Y.-C., On the set-theoretical Yang-Baxter equation, Duke Math. J., 104, 1-18 (2000) · Zbl 0960.16043
[13] Tate, J.; Van den Bergh, M., Homological properties of Sklyanin algebras, Invent. Math., 124, 619-647 (1996) · Zbl 0876.17010
[14] A.P. Veselov, Yang-Baxter maps and integral dynamics, Preprint.; A.P. Veselov, Yang-Baxter maps and integral dynamics, Preprint. · Zbl 1051.81014
[15] Weinstein, A.; Xu, P., Classical solutions of the quantum Yang-Baxter equation, Comm. Math. Phys., 148, 309-343 (1992) · Zbl 0849.17015
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