Pardon me if my terminology is messed up and I must admit that the following question is rather general, but here goes:
Let $A$ be an algebra and $B$ a proper subalgebra of $A$. Suppose that, as $B$-modules, $V$ and $W$ are $B$-isomorphic. Under what conditions (on $V, W$ or $A, B$) can we say that $V$ and $W$ are $A$-isomorphic?
The particular situation I am looking at is that I am given two representations of the braid group, $B_n$ and I know how to show that they are $B_{n-1}$-isomorphic, but I would like to conclude that they are $B_n$-isomorphic.
Thanks!