Power-associative algebras that are train algebras
We investigate the structure of power-associative algebras that are
train algebras. We first show the existence of idempotents, which are all
principal and absolutely primitive. We then study the train equation involving
the Peirce decomposition. When the algebra is finite-dimensional, it turns out
that the dimensions of the Peirce components are invariant and that the upper
bounds for their nil-indexes are reached for some idempotent. Further, locally
train algebras are shown to be train algebras. We then get a complete
description of the set of idempotents by giving their explicit formulas,
including several illustrative examples. Some attention is paid to the Jordan
case, where we discuss conditions forcing power-associative train algebras to
be Jordan algebras. It is also shown that finitely generated Jordan train
algebras are finite-dimensional. For a nth-order Bernstein algebra of period
p, we prove that power-associativity necessitates p=1. In this case, there are
2n-1 possible train equations, which are explicitly described.