I am having trouble understanding the adjoint representation of a Lie algebra in the scope of a very specific example, so I thought physics.SE would be the best place to ask.
Background: A $N \times N$ density matrix $\rho$ contains only $N^2 - 1$ non-redundant real quantities. Therefore, it is convenient to represent it as real vector $d$ (called coherence or pseudospin vector). One possible parameterization [1] of the density matrix is $$\rho = N^{-1}I + \frac{1}{2} \sum_{j=1}^{N^2-1} d_j s_j,$$ where $s_j$ are the generalized Pauli matrices and $I$ is the identity. This parameterization can be plugged into the Liouville-von Neumann equation $\partial_t \rho = \mathcal{L}(\rho)$ and after applying $\mathrm{Tr}\{ \cdot s_i\}$ on both sides, one receives a differential equation for the vector $d$: $\partial_t d = L d$, where $L$ is a real $(N^2-1)\times (N^2-1)$ matrix with the elements $L_{ij} = \mathrm{Tr}\{ \mathcal{L}(s_j) s_i \}$.
Q0: Can the adjoint representation represent elements of a vector space in a different vector space?
Basically, we transform the density matrix (Hermitian, $N \times N$) to a vector (real, $N^2-1$), which is an isomorphism between two vector spaces. However, if I am not mistaken, the adjoint representation considers only one vector space.
Q1: Is (and if yes, how) the matrix $L$ related to the adjoint representation of the $\mathfrak{su}(N)$ algebra?
As far as I understood, the $s_j$ (traceless and Hermitian) span the $\mathfrak{su}(N)$ algebra and can be used to compose both density matrix $\rho$ and Hamiltonian $H$, where the latter enters the Liouvillian $$\mathcal{L}(\rho) = -\mathrm{i}\hbar^{-1} [ H, \rho ].$$ Then, $L$ resembles $\mathrm{ad}_{H} (\rho)$, where the Lie bracket $-\mathrm{i} [\cdot, \cdot]$ is used (I think this bracket must be used with the Hermitian version of the generalized Pauli matrices).
Q2: Does the relation in Q1 change when considering a general Liouvillian?
Of course the commutator term will remain in the Liouvillian, but possibly a dissipation term $\mathcal{G}(\rho)$ (similar to the Lindblad master equation) will be added. The matrix $L$ can be derived for any Liouvillian, but can it still be called adjoint representation?
Q3: How does this all relate to the equation $$\mathrm{Ad}_{\exp(x)} = \exp(\mathrm{ad}_x)?$$
The solution for the vector ODE is $d(t) = \exp({Lt})d(0)$. If $L$ is the adjoint representation of the Liouvillian, then the solution for the original density matrix reads $\rho(t) = \exp(\mathcal{L}t) \rho(0) \exp(-\mathcal{L}t)$. This would make sense, but is it correct? And does it hold for general Liouvillians?
[1] Hioe, F. T., & Eberly, J. H. (1981). $N$-level coherence vector and higher conservation laws in quantum optics and quantum mechanics. Physical Review Letters, 47(12), 838.
Edit #1: Added Q0 for a more basic understanding.
Edit #2: I have asked a separate question on maths.SE regarding the change of basis https://math.stackexchange.com/questions/2682901/matrix-exponential-and-change-of-basis