Deformations and their controlling cohomologies of $\mathcal{O}$-operators

Abstract

$\mathcal{O}$-operators are important in broad areas in mathematics and physics, such as integrable systems, the classical Yang-Baxter equation, pre-Lie algebras and splitting of operads. In this paper, a deformation theory of $\mathcal{O}$-operators is established in consistence with the general principles of deformation theories. On the one hand, $\mathcal{O}$-operators are shown to be characterized as the Maurer-Cartan elements in a suitable graded Lie algebra. A given $\mathcal{O}$-operator gives rise to a differential graded Lie algebra whose Maurer-Cartan elements characterize deformations of the given $\mathcal{O}$-operator. On the other hand, a Lie algebra with a representation is identified from an $\mathcal{O}$-operator $T$ such that the corresponding Chevalley-Eilenberg cohomology controls deformations of $T$, thus can be regarded as an analogue of the Andr\'e-Quillen cohomology for the $\mathcal{O}$-operator. Thereafter, infinitesimal and formal deformations of $\mathcal{O}$-operators are studied. In particular, the notion of Nijenhuis elements is introduced to characterize trivial infinitesimal deformations. Formal deformations and extendibility of order $n$ deformations of an $\mathcal{O}$-operator are also characterized in terms of the new cohomology theory. Applications are given to deformations of Rota-Baxter operators of weight 0 and skew-symmetric $r$-matrices for the classical Yang-Baxter equation. For skew-symmetric $r$-matrices, there is an independent Maurer-Cartan characterization of the deformations as well as an analogue of the Andr\'e-Quillen cohomology controlling the deformations, which turn out to be equivalent to the ones obtained as $\mathcal{O}$-operators associated to the coadjoint representations. Finally, infinitesimal deformations of skew-symmetric $r$-matrices and their corresponding triangular Lie bialgebras are studied.