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Deformations and their controlling cohomologies of O-operators
唐荣 博士(吉林大学)
9月19日上午10:00-11:00  数学楼401报告厅

摘要: 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 O-operators is established in consistence with the general principles of deformation theories. On the one hand, O-operators are shown to be characterized as the Maurer-Cartan elements in a suitable graded Lie algebra. A given O-operator gives rise to a differential graded Lie algebra whose Maurer-Cartan elements characterize deformations of the given O-operator. On the other hand, a Lie algebra with a representation is identified from an 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 O-operator. Thereafter, infinitesimal and formal deformations of 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 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 O-operators associated to the coadjoint representations. Finally, infinitesimal deformations of skew-symmetric r-matrices and their corresponding triangular Lie bialgebras are studied.