I read from a book that the complexification of the Lie algebra $\mathfrak{su}(2)$, noted $\mathfrak{su}(2)_\mathbb{C}$, is in fact the Lie algebra $\mathfrak{sl}(2,\mathbb{C})$, the reason being:
- One basis of $\mathfrak{su}(2)$ is given by
\begin{align}I=\begin{pmatrix}1&0\\0&-1\end{pmatrix}, \quad J=\begin{pmatrix}0&i\\i&0\end{pmatrix},\quad K=\begin{pmatrix}0&1\\-1&0\end{pmatrix}\end{align}.
In the complexification of a real vector space, we autorise multiplication by complex scalars.
Therefore,
\begin{align} \mathfrak{su}(2)_\mathbb{C}&=\mathbb{C}I\oplus\mathbb{C}J\oplus\mathbb{C}K \\ &=\{ \begin{pmatrix} \alpha&\beta\\\gamma&-\alpha \end{pmatrix}: \alpha,\beta,\gamma \in \mathbb{C} \} \\ &=\mathfrak{sl}(2,\mathbb{C}) \end{align}
I was rather convinced by this, but if 2. is true, then consider the element $iJ$ which should be in $\mathfrak{su}(2)_\mathbb{C}$. We don't have \begin{align} (iJ)^\dagger+(iJ)&=0 \end{align}
Thus, it violates the definition of \begin{equation} \mathfrak{su}(2)=\{X \in M_2(\mathbb{C}): \text{Tr}(X)=0, \quad X^\dagger+X=0 \} \end{equation} when we allow multiplication of complex scalars.
And I suppose my comprehension of $\mathfrak{su}(2)_\mathbb{C}$ as "the vector space $\mathfrak{su}(2)$ over the field $\mathbb{C}$" is flawed. Can anyone tell me how should I see it?
Another seemingly more formal definition from the book is
\begin{equation} \mathfrak{su}(2)_\mathbb{C}=\mathfrak{su}(2) \otimes_\mathbb{R} \mathbb{C} \end{equation}
and the multiplication by a complex scalar is given by: for $c,z\in \mathbb{C}$, $X \in \mathfrak{su}(2)$,
\begin{equation}c \cdot (X \otimes_\mathbb{R} z)=X \otimes_\mathbb{R}(cz) \end{equation}
First of all, with this definition, how should we take the complex conjugate of \begin{equation}\begin{pmatrix}0&i\\i&0\end{pmatrix} \otimes_\mathbb{R} i\end{equation} ? Secondly, assuming the book is coherent, how can we consider this tensor product as "autorising multiplication with complex scalars"?
Thank you very much for your time.
$\mathfrak{su}(2)$ is a real subspace of $M_2(\mathbb{C})$, and the action of complex conjugation on $\mathfrak{su}(2) \otimes_{\mathbb{R}} \mathbb{C}$ has nothing to do with the action of either complex conjugation or the conjugate transpose on $M_2(\mathbb{C})$. Rather, it has something to do with the action of complex conjugation on $M_2(\mathbb{C}) \otimes_{\mathbb{R}} \mathbb{C}$.
In general, if $V$ is any real vector space, the elements of $V \otimes_{\mathbb{R}} \mathbb{C}$ look like linear combinations of elements of the form $v \otimes \lambda$, where $v \in V$ and $\lambda \in \mathbb{C}$. The complex conjugate of such an element is $v \otimes \overline{\lambda}$. In particular, the complex conjugate of $v \otimes i$ is $v \otimes (-i) = - v \otimes i$. Note that $v$ is "untouched": only the second term in the tensor product is acted on by complex conjugation.