In the paper Prime Representations from a Homological Perspective. The authors show that $Ext^{1}_{\hat{\mathcal{F}}}(V, V)$ is one-dimensional if and only if $V$ is prime for some modules $V$ of quantum affine algebras.

Therefore $Ext^{1}_{\hat{\mathcal{F}}}(V, V)$ is convient to discribe prime modules. Are there some other situations that $Ext^{1}_A(M, N)$ is used to descibe properties of A-modules $M, N$, where $A$ is some algebra? Are there some references about this? Thank you very much.