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This may actually be a silly question.

Why do we study Lorentz and Poincare Groups?

I have studied a bit of the theory but was wondering what exactly are we talking about when we study the theories, representations and generators of Lorentz and Poincare Groups.

Also, is there a way to represent Lorentz transformations, boosts and translations diagrammatically (explicitly - in the sense that one can look at the diagram and understand what's going on)?