## 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.

### MSC:

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