Explicitly Computing The Action Lie Algebroid for $SL_2(\mathbb{R}) \curvearrowright \mathbb{R}^2$

This is going to be a very classic post, where we’ll chat about a computation my friend Shane did earlier today. His research is largely about symplectic lie algebroids , and recently we’ve been trying to understand the rich connections between poisson geometry, lie algebroids, lie groupoids, and eventually maybe fukaya categories of lie groupoids (following some ideas of Pascaleff ). Shane knows…