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