# Pre-alternative algebras and pre-alternative bialgebras

## TL;DR

This paper introduces pre-alternative algebras, explores their bialgebra theory, and establishes analogies with Lie bialgebras and classical Yang-Baxter equations, expanding algebraic structures.

## Contribution

It defines pre-alternative algebras, develops their bialgebra theory, and introduces $PA$-equations as analogues of the classical Yang-Baxter equation.

## Key findings

- Pre-alternative algebras decompose products into compatible parts.
- Existence of a bialgebra theory for pre-alternative algebras.
- Introduction of $PA$-equations as analogues of Yang-Baxter equations.

## Abstract

We introduce a notion of pre-alternative algebra which may be seen as an alternative algebra whose product can be decomposed into two pieces which are compatible in a certain way. It is also the "alternative" analogue of a dendriform dialgebra or a pre-Lie algebra. The left and right multiplication operators of a pre-alternative algebra give a bimodule structure of the associated alternative algebra. There exists a (coboundary) bialgebra theory for pre-alternative algebras, namely, pre-alternative bialgebras, which exhibits all the familiar properties of the famous Lie bialgebra theory. In particular, a pre-alternative bialgebra is equivalent to a phase space of an alternative algebra and our study leads to what we called $PA$-equations in a pre-alternative algebra, which are analogues of the classical Yang-Baxter equation.