A bialgebra theory of post-Lie algebras via Manin triples and generalized Hessian Lie groups
Dilei Lu, Chengming Bai, Li Guo
TL;DR
The paper develops a bialgebra theory for post-Lie algebras using the Manin triple framework, realized through invariant forms arising from generalized pseudo-Hessian Lie groups. It introduces pp-post-Lie algebras as a partial splitting of post-Lie operations and builds pp-post-Lie bialgebras via matched pairs and invariant bilinear forms, along with a pp-post-CYBE that governs antisymmetric solutions. It further connects these structures to dual $p$-O-operators and pre-pp-post-Lie algebras, providing a coherent pathway from geometric insight to algebraic bialgebra data. The results unify post-Lie theory with generalized Hessian geometry and Rota-Baxter frameworks, offering new tools for constructing and analyzing post-Lie bialgebras with potential links to classical Yang-Baxter-type equations.
Abstract
We develop a bialgebra theory of post-Lie algebras that can be characterized by Manin triples of post-Lie algebras associated to a bilinear form satisfying certain invariant conditions. In the absence of dual representations for adjoint representations of post-Lie algebras, we utilize the geometric interpretation of post-Lie algebras to find the desired invariant condition, by generalizing pseudo-Hessian Lie groups to allow constant torsion for the flat connection. The resulting notion is a generalized pseudo-Hessian post-Lie algebra, which is a post-Lie algebra equipped with a nondegenerate symmetric invariant bilinear form. Moreover, generalized pseudo-Hessian post-Lie algebras are also naturally obtained from quadratic Rota-Baxter Lie algebras of weight one. On the other hand, the notion of partial-pre-post-Lie algebra (pp-post-Lie algebras) is introduced as the algebraic structure underlying generalized pseudo-Hessian post-Lie algebras, by splitting one of the two binary operations of post-Lie algebras. The notion of pp-post-Lie bialgebras is introduced as the equivalent structure of Manin triples of post-Lie algebras associated to a nondegenerate symmetric invariant bilinear form, thereby establishing a bialgebra theory for post-Lie algebras via the Manin triple approach. We also study the related analogs of the classical Yang-Baxter equation, $\mathcal O$-operators and successors for pp-post-Lie algebras. In particular, there is a construction of pp-post-Lie bialgebras from the successors of pp-post-Lie algebras.