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Vector bundles on plane cubic curves and the classical Yang-Baxter equation. (English) Zbl 1317.14074
The classical Yang-Baxter equation is an equation in the germs of meromorphic functions in the $2$-dimensional complex plane taking values on $\mathfrak{g} \otimes\mathfrak{g}$, where $\mathfrak{g}$ is the Lie algebra $\mathfrak{sl}_{n}(\mathbb{C})$. Two solutions related by the group of holomorphic function germs are said to be gauge equivalent. The so-called elliptic and trigonometric solutions wee classified by {\it A. A. Belavin} and {\it V. G. Drinfel’d} [Funct. Anal. Appl. 16, 159--180 (1983; Zbl 0511.22011)], and the rational ones by {\it A. Stolin} [Math. Scand. 69, 57--80 (1991; Zbl 0727.17005); Commun. Math. Phys. 141, 533--548 (1991; Zbl 0736.17006)]. The paper under review studies vector bundles over Weierstrass curves $E=V(wv^{2}-4u^{3}-g_{2}uw^{2}-g_{3}w^{3})\subset\mathbb{P}^{2}$ and attach to any such curve, together with a pair of coprime integers $0<d<n$, a solution $r_{(E,(n,d))}$ of the classical Yang-Baxter equation in a canonical way. This solution is analytic with respect to the parameters $g_{2}$ and $g_{3}$ and different choices along the construction give gauge equivalent solutions (c.f. Theorem $A$). The case of smooth Weierstrass curves is treated in section $3$ and generalized in section 4 to singular ones, through the study of fibrations on elliptic curves. The second part of the paper is devoted to give a description of the solutions for the different types of curves. In section 5, an expression for the elliptic solutions attached to smooth curves is given (c.f. Theorem 5.5). In section 6, the authors treat the case of cuspidal Weierstrass curves ($g_{2}=g_{3}=0$) for which all solutions are rational which, in turn, are the most difficult ones from the point of view of the classification of Belavin-Drinfeld. They describ an algorithm (c.f. Algorithm 6.7) to calculate the corresponding solution. This algorithm encodes a certain recursion (c.f. (5.4)) envolving parabolic subalgebras of $\mathfrak{g}$ which turn out to have a Frobenius structure (c.f. Theorem 7.2). Then, section 8 revisits Stolin’s rational solutions of the equation and section 9 shows that these Stolin solutions correspond (i.e., they are gauge equivalent) to the rational solutions $r_{(E,(n,d))}$, being able to make explicit a “distinguished” class of rational solutions (cf. Theorem $C$).
##### MSC:
 14H60 Vector bundles on curves and their moduli 16T25 Yang-Baxter equations 18E30 Derived categories, triangulated categories 14H70 Relationships of algebraic curves with integrable systems 14D06 Fibrations, degenerations
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