Groups with an involutory antiautomorphism and \(K\)-loops; application to space-time-world and hyperbolic geometry. I
If \((G,\cdot)\) is a group with involutory automorphism \(*\) then under certain conditions the set \(P = \{xx^*\mid x\in G\}\) can be provided with an operation \(+\) such that \((P,+)\) is a \(K\)-loop. \(K\)-loops are generalisations of neardomains. The affine group \(\text{Aff}(F,+)\) of a weak \(K\)-loop \((F,+)\) is defined and it is shown that \(\text{Aff}(F,+)\) can be expressed in terms of certain quasidirect products, such as \((F,+)\rtimes_ Q\text{Aut}(F,+)\). These results are then applied to the special linear group \(SL(2,L)\) where \(L\) is quadratic field extension of a commutative field. For these cases it is shown that the associated loops \((P,+)\) are \(K\)-loops and for \(L = \mathbb{R}\), \(P\) is the unit sphere in the future cone of the associated Minkowski-space-time-world.
scientific article; zbMATH DE number 428634
scientific article