The parabolic geometry generated by the Möbius action of SL(2;ℝ) through the Erlangen Program

Abstract

Inspired by the Erlangen Program of Felix Klein, we have studied the SL(2;ℝ)-action on complex, dual and double numbers focusing mainly on dual numbers. Using the Iwasawa decomposition, we have classified SL(2;ℝ) into three one-parameter subgroups denoted by A,N and K and studied their orbits. We have explored the parabolic geometry associated with dual numbers and studied its SL(2;ℝ)-invariant and other geometric properties along with compactification of the space of dual numbers. We have also studied the Cayley transform and deduced equations representing the parabolic unit circle and disk. Lastly, we have studied the applications of Cayley transform on the A, N and K-orbits using its intertwining property.

Keywords

• matrix lie group
• möbius action
• iwasawa decomposition
• dual numbers
• sl(2;ℝ)
• invariant properties