Deformations and Their Controlling Cohomologies of $${\mathcal{O}}$$ O -Operators

Abstract

$${\mathcal{O}}$$ -operators are important in broad areas of 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 Andre–Quillen cohomology for the $${\mathcal{O}}$$ -operator. Thereafter, linear and formal deformations of $${\mathcal{O}}$$ -operators are studied. In particular, the notion of Nijenhuis elements is introduced to characterize trivial linear 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 Andre–Quillen cohomology, which turn out to have an explicit relationship with the ones obtained as $${\mathcal{O}}$$ -operators associated to the coadjoint representations. Finally, linear deformations of skew-symmetric r-matrices and their corresponding triangular Lie bialgebras are studied.