nLab block decomposition of a category

Idea

When considering a category of representations of some group or algebra, a natural step is to attempt to decompose the category into subcategories (i.e. blocks), which are indecomposable summands. Therefore, any representation can be decomposed uniquely as a direct sum of pieces, one in each block, while any morphism comes as a product of morphisms, one in each block. Then a full understanding of the category is equivalent to a full understanding of all of its blocks. This means that to understand the structure, one only needs to classify these indecomposable summands, which are much simpler. A similar principle applies to the Semiorthogonal Decomposition and the Krull-Schmidt Decomposition.

References

The concept of Block Decomposition of a category, specifically regarding the representation theory of algebraic groups, is usually attributed to:

• Joseph N. Bernstein, Israel M. Gelfand, Sergei I. Gelfand: Category of $\mathfrak{g}$ modules, Funktsionalnyi Analiz i ego Prilozheniya 10 2 (1976) 1-8 [doi:10.1007/BF01077933]

Further developments and discussion:

• Vincent Sécherre, Shaun Stevens: Block Decomposition of the Category of $\ell$-modular smooth representations of $GL_n(F)$ and its inner forms, Annales Scientifiques de l École Normale Supérieure 49 3 (2014) [arxiv.org/abs/1402.5349]