Coherent categorical structures for Lie bialgebras, Manin triples, classical \(r\)-matrices and pre-Lie algebras (Q2155618)

The paper under review presents a categorical structure to each of the classes of Lie bialgebras, Manin triples, classical \(r\)-matrices, \(\mathcal{O}\)-operators and pre-Lie algebras by introducing their morphisms that are compatible with natural correspondences among these classes, so that these correspondences become functors. First,the paper introduces the notion of an endo Lie algebra as a triple \((\mathfrak{g}, [\cdot,\cdot], \phi)\) where \((\mathfrak{g}, [\cdot,\cdot]\) is a Lie algebra and \(\phi:\mathfrak{g}\rightarrow\mathfrak{g}\) is a Lie algebra endomorphism. Then the authors give the equivalent structures of bialgebras, matched pairs and Manin triples for endo Lie algebras. As the next step, the authors extend the classical relations of Lie bialgebras with the classical Yang-Baxter equation as well as classical \(r\)-matrices to the context of endo Lie algebras. This naturally gives rise to a notion of coherent homomorphisms for all \(r\)-matrices, not just the skew-symmetric ones. This notion is shown to be compatible with the coherent homomorphisms of Lie bialgebras, leading to a functor of the corresponding categories. Then the paper introduces the notion of \(\mathcal{O}\)-operators on endo Lie algebras and apply it to define coherent homomorphisms of \(\mathcal{O}\)-operators in such a way that they are compatible with the previously considered coherent homomorphisms of classical \(r\)-matrices. The notion of coherent homomorphisms of \(\mathcal{O}\)-operators is moreover compatible with the natural notion of homomorphism of pre-Lie algebras, giving rise to a pair of adjoint functors between the corresponding two categories. The authors also consider a case where all constructions can be given explicitly, providing natural examples of coherent isomorphisms of Lie bialgebras that are not the previously defined isomorphisms. This further justifies the significance of the coherent homomorphisms of Lie bialgebras introduced in this paper.