quotient groups and SLOCC

Question

I have a math-physics question, which is based on an interest in SLOCC systems for black hole entanglement. The Cartan decomposition of a group $G$ such that $H = G/K$ is such that the derivation or Lie algebraic generators obey $$ [\mathfrak h,~\mathfrak h]~\subset~ \mathfrak h,~[\mathfrak h,~\mathfrak k]~\subset~ \mathfrak k,~[\mathfrak k,~\mathfrak k]~\subset~ \mathfrak h $$ Assume then that we have addition quotient structure with $B~=~G/A$ and $C~=~A/K$. It is then tempting to see relationships between $H~=~G/K$ and the two $B~=~G/A$ and $C~=~A/K$. In particular I am interested in the relationship $$ G/K~\rightarrow~G/A\otimes A/K. $$ The arrow can represent a relationship or for that matter a symmetry breaking process. I have worked out some parts of this, but as a physicist I need a bit of a sanity check on this, as this is a bit outside my area. The mathematical question is what is this relationship?

Answer 1

I am a guest here, or just joined. I am responding to the comment by Studiosus. I am thinking of quotients such as $O(n+2)/O(n)$, then the tensor product, $$ \frac{O(n+2)}{O(n+1)}\otimes\frac{O(n+1)}{O(n)}~=~S^n\otimes S^{n-1}. $$ This is then a Cartesian product of spheres. My observation is that this seems to be a sort of compactification on the manifold $O(n+2)/O(n)$

You are right about getting the $\mathfrak h$ and $\mathfrak k$ switched.