# Left-symmetric bialgebroids and their corresponding Manin triples

## TL;DR

This paper introduces the concept of left-symmetric bialgebroids, explores their relation to Manin triples, and constructs examples from pseudo-Hessian manifolds, expanding the geometric understanding of these algebraic structures.

## Contribution

It defines left-symmetric bialgebroids, establishes their equivalence with Manin triples, and links them to pre-symplectic algebroids and Maurer-Cartan equations.

## Key findings

- Defined left-symmetric bialgebroids and their properties.
- Established the equivalence between Manin triples and bialgebroids.
- Connected the structures to Maurer-Cartan equations and Dirac structures.

## Abstract

In this paper, we introduce the notion of a left-symmetric bialgebroid as a geometric generalization of a left-symmetric bialgebra and construct a left-symmetric bialgebroid from a pseudo-Hessian manifold. We also introduce the notion of a Manin triple for left-symmetric algebroids, which is equivalent to a left-symmetric bialgebroid. The corresponding double structure is a pre-symplectic algebroid rather than a left-symmetric algebroid. In particular, we establish a relation between Maurer-Cartan type equations and Dirac structures of the pre-symplectic algebroid which is the corresponding double structure for a left-symmetric bialgebroid.