  We review the geometry of the space of quantum states
$\mathscr{S}(\mathcal{H})$ of a finite-level quantum system with Hilbert space
$\mathcal{H}$ from a group-theoretical point of view. This space carries two
stratifications generated by the action of two different Lie groups: the
special unitary group $\mathcal{SU}(\mathcal{H})$ and its complexification
$\mathcal{SL}(\mathcal{H})$, the complex special linear group. A stratum of the
stratification generated by $\mathcal{SU}(\mathcal{H})$ is composed of
isospectral states, that is, density operators with the same spectrum, A
stratum of the stratification generated by $\mathcal{SL}(\mathcal{H})$ is
composed of quantum states with the same rank.
  We prove that on every submanifold of isospectral quantum states there is
also a canonical left action of $\mathcal{SL}(\mathcal{H})$ which is related
with the canonical K\"{a}hler structure on isospectral quantum states. The
fundamental vector fields of this $\mathcal{SL}(\mathcal{H})$-action are
divided into Hamiltonian and gradient vector fields. The former give rise to
invertible maps on $\mathscr{S}(\mathcal{H})$ that preserve the von Neumann
entropy and the convex structure of $\mathscr{S}(\mathcal{H})$, while the
latter give rise to invertible maps on $\mathscr{S}(\mathcal{H})$ that preserve
the von Neumann entropy but not the convex structure of
$\mathscr{S}(\mathcal{H})$.
  A similar decomposition is given for the $\mathcal{SL}(\mathcal{H})$-action
generating the stratification of $\mathscr{S}(\mathcal{H})$ into manifolds of
quantum states with the same rank, where gradient vector fields preserve the
rank but do not preserve entropy.
  Some comments on multipartite quantum systems are made. It is proved that the
sets of product states of a multipartite quantum system are homogeneous
manifolds for the action of the complex special linear group associated with
the partition.
