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 $\mathcal O$-operators in the conformal sense. Explicitly, the skew-symmetric part of a conformal linear map T where $T_0=T_\lambda \mid _{\lambda =0}$ is an ${\mathcal {O}}$-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 by-product is that a finite left-symmetric conformal algebra which is a free ${\mathbb {C}}[\partial ]$-module gives a natural ${\mathcal {O}}$-operator, and hence, there is a construction of solutions of conformal classical Yang–Baxter equation and conformal S-equation from the former. Another by-product 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 ${\mathcal {O}}$-operators called Rota–Baxter operators on Lie conformal algebras, and some explicit examples are presented.