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From section 2.4 of von Steinkirk's Introduction to Group Theory for Physicists [PDF]

Defining a set of matrices $T_a$ as $$[T_a]_{bc} \equiv -if_{abc}$$ it is possible to recover (2.1.2): $$[T_a, T_b] = if_{bcd}T_c.$$ The states of the adjoint representation correspond to the generators $\lvert X_a\rangle$. A convenient scalar product is: $$\langle X_a \lvert X_b\rangle = \lambda^{-1} \operatorname{tr}(X_a^\dagger X_b).$$ The action of a generator in a state is: $$\begin{align} X_a\lvert X_b \rangle &= \lvert X_c\rangle\langle X_c \rvert X_a \lvert X_b\rangle \\ &= \lvert X_c\rangle[T_a]_{cb} \\ &= i f_{abc}\lvert X_c\rangle \\ &= \lvert i f_{abc} X_c\rangle \\ &= \lvert[X_a, X_b]\rangle. \end{align}$$

I understand the definition of the adjoint representation. It uses structure constants as matrix components of generators, but I can't understand meaning of the states $|X_{a}\rangle$. What does "correspond" mean? What is the exact definition of $|X_{a}\rangle$?

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It seems that von Steinkirk meant to write$^1$

The states $|X_a\rangle$ of the adjoint representation correspond to the Lie algebra generators $T_a$.

The vector space

$$ V~=~{\rm span}_{\mathbb{R}}\{ |X_a\rangle\mid a=1,\ldots, N \} $$

for the adjoint representation

$$ {\rm ad}: L ~\to~{\rm End}(V) $$

is isomorphic to the Lie algebra itself

$$ L~=~{\rm span}_{\mathbb{R}}\{ T_a \mid a=1,\ldots, N \}. $$

Here the adjoint representation is defined via the Lie bracket

$${\rm ad}(X) |Y\rangle~=~|[X,Y]\rangle, \qquad X,Y~\in~L.$$

See also this related Phys.SE post.

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$^1$ Note that the book contains several typos. Check e.g. the indices in the mentioned formulas!

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