Maximal abelian subalgebras of Lie algebras over $\mathbb{C}$

Let $\mathfrak g$ be the Lie algebra of a compact connected Lie group $G$. Let $\mathfrak g_{\mathbb{C}}$ be the complexification of $\mathfrak g$ and let $\mathfrak h \subset \mathfrak g_{\mathbb{C}}$ be a complex Lie subalgebra satisfying $\mathfrak h + \overline{\mathfrak h} = \mathfrak g_{\mathbb{C}}$. Suppose that $\mathfrak a \subset \mathfrak h$ is a maximal abelian Lie subalgebra of $\mathfrak h$. Does it hold that $\mathfrak a + \overline{\mathfrak a}$ is a maximal abelian subalgebra of $\mathfrak g_{\mathcal C}$?

Edit: A nontrivial example:

Suppose that $G$ is an even dimensional compact Lie group and suppose that is endowed with a left-invariant complex structure(*). Take $\mathfrak h$ as the set of all left-invariant vector fields that annihilates every local holomorphic function on $G$.

(*): This kind of complex structure always exist. In Proposition 2.5 of 1 there is a detailed characterization and in section 5.1 of 2 there is an easy construction.

• Are $\overline{\mathfrak h}$ and $\overline{\mathfrak a}$ just $i\mathfrak h$ and $i\mathfrak a$? – LSpice Apr 4 '18 at 0:33
• Thanks. It seems that the actual question is just set inside $\mathfrak{g}_{\Bbb C}$, so could we not just ask this about any complex algebra, or how does it matter that it comes from a compact real one? In particular, in the title you are interested in Cartan subalgebras, but the question body now is about abelian subalgebras of the complexification -- which, unlike in the compact real case, are not necessarily CSA's. – Torsten Schoeneberg Apr 19 '18 at 14:28
• To clarify the last point, would the (trivial) example $\mathfrak{g} = \mathfrak{su}_2, \mathfrak{h} := \mathfrak{g}_{\Bbb C} \simeq \mathfrak{sl}_2(\Bbb C), \mathfrak{a} = \pmatrix{0 & *\\ 0& 0}$ count, even though $\mathfrak{a}$ is maximal abelian but not a CSA? – Torsten Schoeneberg Apr 19 '18 at 14:29
• Oh but wait ... In my supposed example, I think $\overline{\mathfrak{a}} = \pmatrix{0&0\\*&0}$ according to the complex structure, so actually $\mathfrak{a} +\overline{\mathfrak{a}}$ is not even a Lie subalgebra ... which would be a counterexample, but for a different reason. Sorry if I just add confusion. (It would help to clarify which of the algebras should be considered over which field exactly, cf. my comment to the answer.) – Torsten Schoeneberg Apr 20 '18 at 19:05
• Viewing $\mathfrak{sl}_2(\Bbb C)$ as complexification $(\mathfrak{su}_2)_\Bbb{C}$, the conjugation operation you allude to in your first comment is given by $X \mapsto -\sigma(X)^T$, where $\sigma$ is conjugation on each entry of the matrix. (Accordingly, $\mathfrak{g}$ is the fixed point set of that, the traceless skew-hermitian matrices.) My $\mathfrak{a}$ is a maximal abelian complex subalgebra of the complex algebra $\mathfrak{sl}_2(\Bbb C)$, as asked for in the question. Then $\overline{\mathfrak{a}}$ is as described, and $\mathfrak{a} +\overline{\mathfrak{a}}$ is not even a subalgebra. – Torsten Schoeneberg Apr 21 '18 at 7:06

As discussed in the comments, the answer is no, for a striking reason: In general, $\mathfrak{a} + \overline{\mathfrak{a}}$ is not even a subalgebra.

As example, take $\mathfrak{g} = \mathfrak{su}_2$ which we can e.g. (following Wikipedia) define as the three-dimensional real Lie algebra with basis $u_1, u_2, u_3$ satisfying

$$\left[u_3, u_1\right] = 2u_2, \quad \left[u_1, u_2\right] = 2u_3, \quad \left[u_2, u_3\right] = 2u_1.$$

It is well-known that the complexification $\Bbb C\otimes\mathfrak{g}$ is isomorphic (as complex Lie algebra) to $\mathfrak{sl}_2(\Bbb C)$. Concretely, an isomorphism is given by $$f(-\frac{i}{2}\otimes u_1 -\frac{1}{2}\otimes u_2) = \pmatrix{0&1\\0&0},$$ $$f(-\frac{i}{2}\otimes u_1 +\frac{1}{2}\otimes u_2)=\pmatrix{0&0\\1&0},$$ $$f(-i\otimes u_3) = \pmatrix{1&0\\0&-1}$$

One sees from this that $f$ identifies the conjugation action $a\otimes g \mapsto \bar a\otimes g$ on $\Bbb C\otimes \mathfrak{g}$ with $$\pmatrix{a&b\\c&-a} \mapsto \pmatrix{-\bar a&-\bar c\\-\bar b&\bar a}$$ on $\mathfrak{sl}_2(\Bbb C)$ (and $\mathfrak{g}$ with the fixed point set of that, the traceless skew-hermitian matrices, in particular $u_1 = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, \quad u_2 = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \quad u_3 = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$ ).

But then if we choose $\mathfrak{h} = \mathfrak{g}_{\Bbb C}$, and $\mathfrak{a}$ as the one-dimensional algebra generated by $\pmatrix{0&1\\0&0}$ (which is maximal abelian in $\mathfrak{h}$), we have $\overline{\mathfrak{a}} = \pmatrix{0&0\\*&0}$, hence $$\mathfrak{a} + \overline{\mathfrak{a}} = \lbrace \pmatrix{0&b\\c&0}: b,c\in \Bbb C\rbrace$$ which is not closed under the Lie bracket, hence no Lie subalgebra of $\mathfrak{g}_{\Bbb C}$.

NB. If (in the special case $\mathfrak{h} = \mathfrak{g}_{\Bbb C}$) $\mathfrak{a}$ is chosen as a Cartan subalgebra of $\mathfrak{g}_{\Bbb C}$, in some low-dimensional examples it seems to me that $\overline{\mathfrak{a}} = \mathfrak{a}$ and hence the assertion is true. Maybe one can go from there to a case of more general $\mathfrak{h}$. However, I am not even sure if/why this should hold in the special case. (I've asked this as a new question MSE/2756715: Are all CSA's in $\mathfrak{g}_{\Bbb C}$ invariant under the conjugation action?)

I am sorry to tell I believe the answer is No.

Consider $\mathfrak g = \mathfrak{su}(2)$ and its complexification $\mathfrak g_{\mathbb C} = \mathfrak{sl}(2,\mathbb C)$, and $\mathfrak h= \mathfrak{sl}(2,\mathbb R)$. Now look at $\mathfrak a=\mathfrak{so}(2)$, the set of antisymmetric matrices in $\mathfrak g$. Then $\mathfrak a+\bar{\mathfrak a}$ is not maximal Abelian in $\mathfrak g_\mathbb C$, because it is properly contained in $\mathfrak a +\bar{\mathfrak a} + \mathbb C\begin{pmatrix} i \\ 1 \\ -1 \\ -i \end{pmatrix}$. There are more example like this.

• It looked like a lot of formatting got destroyed in your original post, so I edited it, hopefully without changing meaning. The only semantic change I made was adding the missing word 'Abelian' to "$\mathfrak a + \bar{\mathfrak a}$ is not maximal in $\mathfrak g_{\mathbb C}$". – LSpice Apr 20 '18 at 17:08
• Hmm ... I had assumed $\mathfrak{h}$ and then $\mathfrak{a}$ had to be complex subalgebras of $\mathfrak{g}_{\Bbb C}$, which yours are are not. But then I fail to see how $\mathfrak{a} + \overline{\mathfrak{a}}$ is a complex subalgebra of $\mathfrak{g}_{\Bbb C}$ at all; in your example, is the conjugation action on $\mathfrak{a}$ not just multiplication with $-1$, and hence $\mathfrak{a} = \overline{\mathfrak{a}}$? Or are we talking about $\mathfrak{g}_{\Bbb C}$ and everything inside it as real algebras all the time? Maybe the OP can clarify. – Torsten Schoeneberg Apr 20 '18 at 18:40
• In fact, I forgot to make it explicit that $\mathfrak h$ has to be a complex subalgebra of $\mathfrak{su}(2) \otimes \mathbb{C}$. Anyway, in this example, it does not hold that $\mathfrak{h} + \overline{\mathfrak{h}} = \mathfrak g_{\mathbb{C}}$. – Max Reinhold Jahnke Apr 21 '18 at 0:24