Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O}N$-structures

Abstract

In this paper, we first study infinitesimal deformations of a Lie conformal algebra and a Lie conformal algebra with a module (called an $\mathsf{LCMod}$ pair), which lead to the notions of Nijenhuis operator on the Lie conformal algebra and Nijenhuis structure on the $\mathsf{LCMod}$ pair, respectively. Then by adding compatibility conditions between Nijenhuis structures and $\mathcal{O}$-operators, we introduce the notion of an $\mathcal{O}N$-structure on an $\mathsf{LCMod}$ pair and show that an $\mathcal{O}N$-structure gives rise to a hierarchy of pairwise compatible $\mathcal{O}$-operators. In particular, we show that compatible $\mathcal{O}$-operators on a Lie conformal algebra can be characterized by Nijenhuis operators on Lie conformal algebras. Finally, we introduce the notions of conformal $r$-matrix-Nijenhuis structure and symplectic-Nijenhuis structure on the Lie conformal algebra and study their relations.

26 pages

References in corpus (3)

• Field Algebras
• A Unified Algebraic Approach to Classical Yang-Baxter Equation
• Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation

Cited by in corpus (1)

• Rota-Baxter operators on $\mathrm{Cur}({\mathrm{sl}}_2(\mathbb{C}))$