In non-semisimple category, for example the category of representations of a quantum affine algebra $U_q(\hat{g})$, where $g$ is a simple Lie algebra over $\mathbb{C}$. What are the differences between the concepts submodules and subfactors? Thank you very much.


A subfactor is a submodule of the semisimplification. Any module has a Jordan--Holder decomposition (e.g. increasing chain of submodules with each successive quotient simple); the semisimplification is the semisimple module formed by taking the direct sum of these simple quotients. Not every subfactor needs to be a subquotient though: for example, let $\Bbb{Z}$ act on $\Bbb{C}^2$ via $n\mapsto\begin{pmatrix}1 & n\\ 0 & 1\end{pmatrix}$. There's a 1-dimensional submodule $M$ (generated by $\begin{pmatrix} 1\\ 0\end{pmatrix}$), so $\Bbb{C}^2$ has Jordan--Holder series $0\subset M\subset\Bbb{C}^2$ with the quotient $Q=\Bbb{C}^2/M$ simple. But $\Bbb{C}^2\not\simeq M\oplus Q$ since the map $\Bbb{C}^2\rightarrow Q$ doesn't admit a splitting.

  • $\begingroup$ Thank you very much. Let M be a module. Is it possible that there is a subfactor N of M but N is not a submodule of M. $\endgroup$ – LJR Sep 29 '16 at 3:43
  • $\begingroup$ In your example, Q is a subfactor of $\mathbb{C}^2$ but not a submodule. Am I correct? $\endgroup$ – LJR Sep 29 '16 at 5:01
  • $\begingroup$ Yes, that's right. $\endgroup$ – PL. Sep 29 '16 at 9:43

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