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.

中文翻译：

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保形经典杨-巴克斯特方程、S 方程和 $${\mathcal {O}}$$O-operators

在李保形双代数和左对称保形双代数的研究中自然会出现保形经典杨-巴克斯特方程和S-方程。在本文中，它们被解释为一种运算符，即保形意义上的 $$\mathcal O$$-operators。明确地，保形线性映射 T 的偏斜对称部分，其中 $$T_0=T_\lambda \mid _{\lambda =0}$$ 是保形中的 $${\mathcal {O}}$$-operator sense 是保形经典 Yang-Baxter 方程的偏对称解，而对称部分是保形 S 方程的对称解。一个副产品是有限左对称保形代数是一个免费的 $${\mathbb {C}}[\partial ]$$-module 给出一个自然的 $${\mathcal {O}}$$-operator ， 因此，有一个共形经典杨-巴克斯特方程的解的构造和前者的共形 S 方程的解。另一个副产品是这两个方程的非退化解分别对应于李保形代数和左对称保形代数的 2-cocycles。我们还进一步研究了一类特殊的 $${\mathcal {O}}$$-operators，称为 Lie 保形代数上的 Rota–Baxter 算子，并给出了一些明确的例子。