My post here concerns the concept of Lagrangian subgroup for a non-abelian Lie group, such as a semi-simple non-abelian Lie group for gauge theory.

The symplectic form $\omega$  is given by (with the restricted $a_{\parallel,I}$ on ${\partial \mathcal M}$ ),
$$
\omega=\frac{K_{IJ}}{4\pi} \int_{ \mathcal M}  (\delta a_{\parallel,I}) \wedge d (\delta a_{\parallel,J}).
$$
for a varyation of the differential of this 1-form on a abelian Chern-Simons action $\delta S_{bluk}$.

For the well-known abelian gauge group such as the bulk gauge group $\text{U(1)}^N  \cong \mathbb{T}_\Lambda$ as the torus, is the quotient space of $N$-dimensional vector space $\bf{V}$ by a subgroup $\Lambda \cong \mathbb Z^N$.
Locally the gauge field $a$ is a 1-form,
which has values in the Lie algebra of $\mathbb{T}_\Lambda$, we can denote this Lie algebra $\mathbf{t}_\Lambda$ as the vector space $\mathbf{t}_\Lambda =\Lambda \otimes \mathbb{R}$.


>For topological gapped boundary,
$a_{\parallel,I} \quad$  lies in a Lagrangian subspace of $\mathbf{t}_\Lambda$ implies that the boundary gauge group ($ \equiv \mathbb{T}_{\Lambda_{0}}$)
is a Lagrangian subgroup. 

We can rephrase it in terms of the exact sequence
for the vector space of Abelian group 
$$\Lambda \cong \mathbb Z^N$$ 
and its subgroup $\Lambda_0$:

These form an exact sequence:
$$
0 \to \Lambda_0 \overset{\mathbf{h}}{\to} \Lambda \to \Lambda/\Lambda_0 \to 0.
$$
Here $0$ means the trivial Abelian group with only the identity, or the zero-dimensional vector space.
Here $0$ means the trivial zero-dimensional vector space and 
$\mathbf{h}$ is an injective map from $\Lambda_0$ to $\Lambda$. 

We can also rephrase it in terms of the exact sequence
for the vector space of Lie algebra by 

$$0 \to \mathbf{t}_{(\Lambda/\Lambda_0)}^* \to  \mathbf{t}_{\Lambda}^* \to \mathbf{t}_{\Lambda_0}^* \to  0.$$

> **My Question**: For topological gapped boundary, do we have a direct generalization such that the
$a_{\parallel,I}$ lies in a **Lagrangian subspace** of $\mathbf{t}_\Lambda$ implies that the **boundary gauge group**
is still a **Lagrangian subgroup**? How do we define the Lagrangian subgroup of a nonabelian Lie group? 

- References along this direction is very welcome.