Why has the adjoint representation a higher dimension than the basis matrices it acts on? for example here(adsbygoogle = window.adsbygoogle || []).push({});

Why is e_1 two dim and ad(e_1) four dim?

Isn't ad(X) Y a simple matrix multiplication here? But then multiplying 4x4 with 2x2 matrices, what does it mean?

thanks

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# Adjoint representation

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