Pre-alternative algebras and pre-alternative bialgebras

Details Pre-alternative algebras and pre-alternative bialgebras Pre-alternative algebras and pre-alternative bialgebras

2009-07-20

arxiv.org/abs/0907.3391v1

Pacific Journal of Mathematics 248 (2010) 355-391

Physics

Mathematical Physics

34 pages

Scientific paper

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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