Geometry of the non-compact G(2)

Abstract

Geometrical applications of the non-compact form of Cartan’s exceptional Lie group G(2) are considered. This group comprises specific rotations of 7-dimensional Minkowski-like space with three extra time-like coordinates and in some limiting cases imitates standard Poincare transformations. In this model space–time translations are non-commutative and are represented by the rotations towards the extra time-like coordinates. The second order Casimir element of non-compact G(2) and its expression by the Casimir operators of the Lorentz and Poincare groups are found.