Two models of non-Euclidean spaces generated by associative algebras. (English) Zbl 1235.53015

The author considers the real unital algebra \(\mathfrak A\) with basis elements \(1,e_1,e_2\) and defining relations \(e_1^2=1\), \(e_1e_2=e_2=-e_2e_1\), \(e_2^2=0\). This is—up to isomorphism—the algebra of real upper triangular \(2\times 2\) matrices (which is also known under the name of real ternions). From a geometric point of view the non-units of \(\mathfrak A\) comprise two planes through \(0\), their intersection being the Jacobson radical of the algebra. So it is quite natural to endow \(A\) with a degenerate pseudo-Euclidean dot product with signature \({+}{-}0\). The isometry group, the geometry on a sphere of this space (which is a hyperbolic cylinder) and a projective conformal model of this sphere are exhibited.


53A35 Non-Euclidean differential geometry
53A20 Projective differential geometry
53A30 Conformal differential geometry (MSC2010)
55R10 Fiber bundles in algebraic topology