What are the differences between submodules and subfactors.

Question

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.

Answer 1

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.