Question on quotient groups and SLOCC

Question

I have a math-physics question, which is based on an interest in stochastic local operator & classical communications (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. The mathematical question is what is this relationship?

Answer 1

I gather your unfortunate edit, which further confused things, to the point of intractability, came from verbatim reproduction of eqn (3.23) of the Duff et al paper on Stochastic Local Operations and Classical Communication. One of about a dozen problems with your malformed question is that you used the wrong coset space to be described by the Cartan pair you are writing down, and, moreover, as ACuriousMind points out, these spaces are rarely groups. Since the Duff et al paper presumes standard practical group theory covered in introductory courses in particle theory, I will eschew mathematese, (which can only lead to grief and chaos, as here, by its systematic misuse) and stick to bare-bones concepts and terms familiar to the random particle theorist in the street.

But, naturally, I still do not know what this magnificently sybilline tensor product in the punchline of your question is supposed to mean, here--I'll just assume you are interested in sequential SSB and its projective coordinate parameterization, the σ -models involved. You might benefit by searching on coset spaces on this site, e.g. from this question, 110148.

Given a group $G$ and a subgroup of it $K$ to which it is spontaneously broken, the broken generators ("axials" in the chiral symmetry breaking paradigm of low energy QCD, $SU(2)\times SU(2)/SU(2)_{isospin}$) is the coset space $H = G/K$. The Generators of G than break up into the unbroken ones, $\mathfrak k$, (isospin), and the broken ones, $\mathfrak h$, parameterized by the goldstone/pions serving as projective coordinates of that manifold (In QCD this is just $S^3$): $$ [\mathfrak h,~\mathfrak h]~\subset~ \mathfrak k,\qquad [\mathfrak h,~\mathfrak k]~\subset~ \mathfrak h, \qquad [\mathfrak k,~\mathfrak k]~\subset~ \mathfrak k. $$ In your question, eager to follow the notation of Duff et al, you reversed the role of broken vs unbroken generators. Remember the unbroken generators (isospin) close to a subalgebra, but the broken ones (axials) transform by the $\mathfrak k$ as isomultiplets and Lie-commutator-close to vectors $\mathfrak k$!

Now suppose you wish to keep breaking spontaneously (not explicitly!), say break $K$ to $A$. (I switched your nomenclature for continuity, not slick paltering.) Suppose $K$ being isospin, you break it down to $A= U(1)$, the third component thereof, $I_3$. Now two generators of $K$ will develop their two goldstons ($g_{\pm}$) parameterizing $K/A \sim S^2$, and they will be rotated to each other by $I_3$, the only unbroken generator of the original 6, $SU(2)\times SU(2) / U(1)\sim G/A$. (Unphysically: QCD doesn't do that! In general, $\mathfrak k$: $\mathfrak a$ unbroken and $\mathfrak b$ broken, satisfying analogous Lie algebra relationships to $\mathfrak k$ and $\mathfrak h$, above).

Relationships such as your (relabeled) $$ G/A \qquad \leftrightarrow~G/K\otimes K/A, $$ representing $S^3 \otimes S^2$ (a hypersphere of pions tensored to a beach-ball of g s) may be badly misleading, as K is broken, so your former hypersphere is not a hypersphere anymore---it's not a symmetric space, having melted in the second breaking to a nightmare out of Dali's brushes. Consider $[\mathfrak h,~\mathfrak b]~\not\subset~ \mathfrak h$; the closest symmetric space is $SO(4)/(SO(2)\times SO(2))$.

I have no idea how to represent the resulting 5-dim manifold, but clearly if the two breakings ocurred at vastly different SSB scales, there might be a perturbative theory to parameterize it tastefully to evoke the r.h.side tensor product.

I would tinker with simple models first, long before I got into cramming heavy-duty math books and misusing terminology, though. Your title, as you saw, sends readers off on a completely unintended tangent.