Solutions of the classical Yang-Baxter equation for simple superalgebras
The equation \([X^{12}(u),X^{13}(u+v)]+[X^{12}(u),X^{23}(v)]+[X^{13}(u+v),X^ {23}(v)]=0,\) where \({\mathfrak G}\) is a Lie (super)algebra and \(X^{ij}:\quad {\mathbb{C}}\to U({\mathfrak G})^ 3\) is called the classical Yang-Baxter equation. Here \(U({\mathfrak G})\) is the universal enveloping algebra of the Lie (super)algebra \({\mathfrak G}\) and \(X^{12}({\mathbb{C}})\subset {\mathfrak G}\otimes {\mathfrak G}\otimes 1, X^{13}({\mathbb{C}})\subset {\mathfrak G}\otimes 1\otimes {\mathfrak G}, X^{23}({\mathbb{C}})\subset 1\otimes {\mathfrak G}\otimes.\) In this paper all nondegenerate solutions of the classical Yang-Baxter equation for simple finite-dimensional Lie superalgebras which are either elliptic or trigonometric or constant functions are found. These solutions are given to within equivalence concerning the following transformations: \(1)\quad X(u)\to X(u)+r,\) where \({\mathfrak h}\) is a Lie subalgebra of \({\mathfrak G}\), \(r\in {\mathfrak h}\otimes {\mathfrak h}, r^{12}=-r^{21}, [r^{12},r^{23}]+[r^{12},r^{13}]+[r^{13},r^{23}]=0; 2)\quad X(u)\to cX(u),\) where \(c\in {\mathbb{C}}\backslash 0\); \(3)\quad X(u_ 1-u- 2)\to(\phi(u_ 1)\otimes \phi(u_ 2))X(u_ 1-u_ 2),\) where \(\phi\) (u)\(\in Aut {\mathfrak G}\).
scientific article; zbMATH DE number 3853390