Coherent categorical structures for Lie bialgebras, Manin triples, classical $r$-matrices and pre-Lie algebras

TL;DR

This paper develops a categorical framework for Lie bialgebras, Manin triples, classical r-matrices, and pre-Lie algebras by introducing coherent homomorphisms, enhancing understanding of their interrelations.

Contribution

It introduces the concept of coherent homomorphisms for these algebraic structures, providing a unified categorical perspective and generalizing classical notions to endo Lie algebras.

Findings

Categories of these structures are established with coherent homomorphisms.

The framework generalizes classical notions to endo Lie algebras.

Compatibility with existing correspondences and pre-Lie algebras is demonstrated.

Abstract

The broadly applied notions of Lie bialgebras, Manin triples, classical $r$-matrices and $\mathcal{O}$-operators of Lie algebras owe their importance to the close relationship among them. Yet these notions and their correspondences are mostly understood as classes of objects and maps among the classes. To gain categorical insight, this paper introduces, for each of the classes, a notion of homomorphisms, uniformly called coherent homomorphisms, so that the classes of objects become categories and the maps among the classes become functors or category equivalences. For this purpose, we start with the notion of an endo Lie algebra, consisting of a Lie algebra equipped with a Lie algebra endomorphism. We then generalize the above classical notions for Lie algebras to endo Lie algebras. As a result, we obtain the notion of coherent endomorphisms for each of the classes, which then generalizes to the notion of coherent homomorphisms by a polarization process. The coherent homomorphisms are compatible with the correspondences among the various constructions, as well as with the category of pre-Lie algebras.