Physics student here. In my textbook on group theory, there was a matrix $$h_1= \begin{pmatrix} 1/2 & 0 & 0\\ 0 & -1/2 & 0\\ 0 & 0 & 0 \end{pmatrix}$$ Then the author define the operator $H_1=D(h_1)$ where D is an irreducible representation of the Lie algebra of SU(3). I can't understand why they defined it as such. I know that the matrix $h_1$ is the 3rd generator of the Lie algebra of SU(3), that is $\frac{1}{2}\lambda_3 $. Thanks.
2026-02-27 16:16:01.1772208961
Confusion about operators and representations
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What you have written down is the triplet (fundamental, defining) SU(3) representation of $h_1$, the first Cartan subalgebra generator.
It is called the isospin generator for the triplet in physics-speak. I'm not sure what troubles you. You should be able to write the octet (adjoint) representation of the same generator, as well, given the structure constants of the su(3) Lie algebra.