Deformations and their controlling cohomologies of \(\mathcal{O}\)-operators.(English)Zbl 1440.17015
Summary: \(\mathcal{O}\)-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 André-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 André-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.
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 André-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.
MSC:
| 17B55 | Homological methods in Lie (super)algebras |
| 16T25 | Yang-Baxter equations |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 17B38 | Yang-Baxter equations and Rota-Baxter operators |
| 17B40 | Automorphisms, derivations, other operators for Lie algebras and super algebras |
| 17B56 | Cohomology of Lie (super)algebras |
| 17B62 | Lie bialgebras; Lie coalgebras |
Cite
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