nLab finite abelian category

Definition

Definition

Let $k$ be a field, and let $\mathcal{C}$ be a $k$-linear abelian category (i.e. one whose Ab-enrichment is lifted to a Vect-enrichment). Then $\mathcal{C}$ is called [Etingof & Ostrik 2003 p. 3] finite (over $k$) if

• For any two objects $a$, $b$ of $C$, the hom-object ($k$-vector space) $\hom(a, b)$ has finite dimension;

• Each object $a$ is of finite length;

• There are only finitely many simple objects in $C$, and each of them admits a projective presentation.

Theorem (Deligne)

For any finite abelian category $C$, there exists a finite-dimensional $k$-algebra $A$ and an $k$-linear equivalence between $C$ and $A$-$Mod_{fd}$, the category of modules over $A$ that are finite-dimensional as vector spaces over $k$.

Related concepts

• semisimple category

• finite-dimensional vector space

• finite-dimensional Hilbert space

References

• Pavel Etingof, Victor Ostrik, Finite Tensor Categories [arXiv:math/0301027]

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, chapter 6 of Tensor categories, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society (2015) [pdf]