The Geometry of the Projective Action of $\text{SL}(3,\mathbb{R})$ from the Erlangen Perspective

Debapriya Biswas; Ipsita Rajwar

doi:10.52737/18291163-2024.16.1-1-28

Authors

Debapriya Biswas Indian Institute of Technology Kharagpur
Ipsita Rajwar Indian Institute of Technology Kharagpur

DOI:

https://doi.org/10.52737/18291163-2024.16.1-1-28

Keywords:

Lie Group SL(3,R), Homogeneous Space, Conics, Exponential Map, Iwasawa Decomposition

Abstract

In this paper, we have investigated the projective action of the Lie group $\text{SL}(3,\mathbb{R})$ on the homogeneous space $\mathbb{RP}^2$. In particular, we have studied the action of the subgroups of $\text{SL}(3,\mathbb{R})$ on the non-degenerate conics in the space $\mathbb{RP}^2$. Using the Iwasawa decomposition of $\text{SL}(2,\mathbb{R})$, we demonstrate that the isotropy subgroup of the projective unit circle is isomorphic to $\text{PSL}(2,\mathbb{R})$ under certain conditions.

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