$\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras

TL;DR

This paper explores $ abla$-operators and Nijenhuis operators in associative conformal algebras, introducing new algebraic structures and cohomology theories to deepen understanding of their properties and relationships.

Contribution

It introduces twisted Rota-Baxter operators, conformal NS-algebras, and a conformal Nijenhuis operator, expanding the algebraic framework of associative conformal algebras.

Findings

Twisted Rota-Baxter operators induce conformal NS-algebras.

$ abla$-operators are characterized with key properties.

A cohomology theory for $ abla$-operators is constructed.

Abstract

We study $\mathcal{O}$-operators of associative conformal algebras with respect to conformal bimodules. As natural generalizations of $\mathcal{O}$-operators and dendriform conformal algebras, we introduce the notions of twisted Rota-Baxter operators and conformal NS-algebras. We show that twisted Rota-Baxter operators give rise to conformal NS-algebras, the same as $\mathcal{O}$-operators induce dendriform conformal algebras. And we introduce a conformal analog of associative Nijenhius operators and enumerate main properties. By using derived bracket construction of Kosmann-Schwarzbach and a method of Uchino, we obtain a graded Lie algebra whose Maurer-Cartan elements are given by $\mathcal{O}$-operators. This allows us to construct cohomology of $\mathcal{O}$-operators. This cohomology can be seen as the Hochschild cohomology of an associative conformal algebra with coefficients in a suitable conformal bimodule.