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In B.C.Hall mathematics book on Lie Groups the Lie algebra is by definition closed under the Lie bracket operation, that is $[X, Y]\in \mathfrak g$ for every $X,Y \in \mathfrak g$.

The $\mathfrak su(2)$ algebra is defined as the $2 \times 2$ skew-Hermitian matrices with trace zero, and it's easy to see that $[X, Y]$ is skew-Hermitian, so $[X, Y] \in \mathfrak su(2)$.

However, in A.Zee Physics book on Group the $\mathfrak su(2)$ algebra is defined as the $2 \times 2$ Hermitian matrices with trace zero , and it's easy to see that $[X, Y]$ is skew-Hermitian, not Herimitian, so in this way $\mathfrak su(2)$ is not closed under the standard Lie bracket.

So to preserve the closure, it seems to me that the Lie bracket should be redefined as $[X,Y]\equiv -i(XY-YX)$. Since that is not mentioned at all in the A.Zee book I'm wondering if my arguments are correct.

Qmechanic
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Andrea
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    Related: https://physics.stackexchange.com/q/321230/291677 – Quantum Mechanic Sep 27 '21 at 14:23
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    Yes, you are right. The reason physicists prefer this convention is that Hermitian operators/matrices appear naturally in quantum physics and have physical meaning. You will see physicist write Lie Group elements (often appear in the form of unitary representations in quantum physics) as $g=e^{iX}$, with Hermitian $X$, rather than the standard math convention of $g=e^{X}$. – Heidar Sep 27 '21 at 14:26

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