Infinite-dimensional Lie bialgebras via affinization of perm bialgebras and pre-Lie bialgebras

Abstract

It is known that the operads of perm algebras and pre-Lie algebras are the Koszul dual each other and hence there is a Lie algebra structure on the tensor product of a perm algebra and a pre-Lie algebra. Conversely, we construct a special perm algebra structure and a special pre-Lie algebra structure on the vector space of Laurent polynomials such that the tensor product with a pre-Lie algebra and a perm algebra being a Lie algebra structure characterizes the pre-Lie algebra and the perm algebra respectively. This is called the affinization of a pre-Lie algebra and a perm algebra respectively. Furthermore we extend such correspondences to the context of bialgebras, that is, there is a bialgebra structure for a perm algebra or a pre-Lie algebra which could be characterized by the fact that its affinization by a quadratic pre-Lie algebra or a quadratic perm algebra respectively gives an infinite-dimensional Lie bialgebra. In the case of perm algebras, the corresponding bialgebra structure is called a perm bialgebra, which can be independently characterized by a Manin triple of perm algebras as well as a matched pair of perm algebras. The notion of the perm Yang-Baxter equation is introduced, whose symmetric solutions give rise to perm bialgebras. There is a correspondence between symmetric solutions of the perm Yang-Baxter equation in perm algebras and certain skew-symmetric solutions of the classical Yang-Baxter equation in the infinite-dimensional Lie algebras induced from the perm algebras. In the case of pre-Lie algebras, the corresponding bialgebra structure is a pre-Lie bialgebra which is well-constructed. The similar correspondences for the related structures are given.