Non-semisimple representations refer to representations of algebraic structures, such as groups or algebras, that cannot be decomposed into a direct sum of simple representations. In simpler terms, these representations contain sub-representations that are not simple, meaning they have further internal structure. This complexity can lead to challenges in understanding the overall behavior of the representation. In the context of representation theory, non-semisimple representations often arise in the study of Lie algebras or finite groups where the underlying structure is more intricate. These representations can exhibit phenomena like reducibility, where a representation can be broken down into smaller parts, complicating the analysis and application of the theory.