Conformal classical Yang-Baxter equation, $S$-equation and $\mathcal{O}$-operators

Abstract

Conformal classical Yang-Baxter equation and S-equation naturally appear in the study of Lie conformal bialgebras and left-symmetric conformal bialgebras. In this paper, they are interpreted in terms of a kind of operators, namely, O-operators in the conformal sense. Explicitly, the skew-symmetric part of a conformal linear map T where T0 = T λ | λ=0 is an O-operator in the conformal sense is a skew-symmetric solution of conformal classical Yang-Baxter equation, whereas the symmetric part is a symmetric solution of conformal S-equation. One byproduct is that a finite left-symmetric conformal algebra which is a free C[∂]-module gives a natural O-operator and hence there is a construction of solutions of conformal classical Yang-Baxter equation and conformal S-equation from the former. Another byproduct is that the non-degenerate solutions of these two equations correspond to 2-cocycles of Lie conformal algebras and left-symmetric conformal algebras respectively. We also give a further study on a special class of O-operators called Rota-Baxter operators on Lie conformal algebras and some explicit examples are presented.