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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!

  • 0
    @ Zhen Lin Why $\mathbb{C}^n$ and $\mathbb{C}^m$ are $\mathbb{Q}$-isomorphic?2016-10-26

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