# Pre-alternative algebras and pre-alternative bialgebras

Xiang Ni, Chengming Bai

Arxiv ID: 0907.3391•Last updated: 9/20/2022

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.

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