Alternative algebras with the hyperbolic property

Details Alternative algebras with the hyperbolic property Alternative algebras with the hyperbolic property

2008-10-24

arxiv.org/abs/0810.4544v6

Mathematics

Group Theory

Third author's Ph.D. (parcial); generalization of those results for the non-associative case, 9 pps. Conf.: Geometry, Topology

Scientific paper

We investigate the structure of an alternative finite dimensional $\Q$-algebra $\mathfrak{A}$ subject to the condition that for a $\Z$-order $\Gamma \subset \mathfrak{A}$, and thus for every $\Z$-order of $\mathfrak{A}$, the loop of units of $\U (\Gamma)$ does not contain a free abelian subgroup of rank two. In particular, we prove that the radical of such an algebra associates with the whole algebra. We also classify $RA$-loops $L$ for which $\mathbb{Z}L$ has this property. The classification for group rings is still an open problem.

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