additive and abelian categories
(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
left/right exact functor
A semisimple category is a category in which each object is a direct sum of finitely many simple objects, and all such direct sums exist.
(semisimple abelian category)
An abelian category is called semisimple if every object is a semisimple object, hence a direct sum of finitely many simple objects.
Alternatively, at least over an algebraically closed ground field $k$:
(semisimple linear category)
A linear category (that is, a category enriched over Vect, cf. tensor category) is called semisimple if:
it has finite biproducts
(usually called ‘direct sums’),
(so we say that it ‘has subobjects’ or, perhaps better, ‘has retracts’),
there exist objects $X_i$ labeled by an index set $I$ such that
for any pair $i, j$ of indices we have an isomorphism
where $\delta$ denotes the Kronecker delta and $k$ the ground field
(such objects are called simple),
for any pair of objects $V,\,W$, the natural composition map
is an isomorphism.
(If the field $k$ is not algebraically closed, a simple object $X$ can have a finite-dimensional division algebra other than $k$ itself as its endomorphism algebra.)
(e.g. Müger (2003, p. 6))
(direct sum decomposition into simple objects)
Definition implies that every object $V$ is a direct sum of simple objects $X_i$.
The third item of the definition is equivalent to stipulating that the vector space $Hom(X_i, V)$ is in canonical duality with the vector space $Hom(V, X_i)$. Indeed, we have:
a pairing
where on the right we used (1),
a copairing
where on the right we used (2),
and one checks that these satisfy the triangle identities and as such exhibit dual vector spaces.
Hence if we choose a linear basis
for each vector space $Hom(X_i, V)$, we get a corresponding dual basis
satisfying
This says precisely that $V$ has been expressed as a direct sum of the $X_i$.
Def. (of semisimple linear categories) does not use the concept of abelian category. This is because the concepts that one thinks about with abelian categories such as kernels and cokernels do not play an important conceptual role in semisimple categories, being replaced by the more important concepts of biproduct and retract. Hence it is best to give a streamlined definition from first principles without going through the language of abelian categories which would have muddied the waters.
For a category to be semisimple, it needs to have a certain directional symmetry in its hom-sets, namely that $Hom(V, W)$ must at least have the same dimension as $Hom(W,V)$. This is the easiest way to check if a category will fail to be semisimple. For instance, the category $Rep(A)$ of representations of an algebra $A$ will rarely be semisimple, precisely because there is no relation between $Hom(V, W)$ and $Hom(W,V)$ in general. Again, this can be traced back to the original algebra $A$ not having any ‘symmetry’ like the inverse operation in a group.
As far as ‘duality’ on the hom-sets is concerned, one might have a $S \colon C \rightarrow C$ from the category to itself with the property that there are canonical isomorphisms
where “$\vee$” denotes the ordinary linear dual of a vector space. Such a functor is called a Serre functor in algebraic geometry, and indeed there is precisely such a functor on the derived category of coherent sheaves on a complex manifold — it is given by tensoring with the canonical line bundle.
For 2-Hilbert spaces, there is an antilinear $*$-operation on the hom-sets $* : Hom(V, W) \rightarrow Hom(W,V)$. The presence of this duality in fact forces the category to be semisimple (this comes down to the fact that a finite-dimensional $*$-algebra, such as the hom’s between a bunch of objects in the category, must be a full matrix algebra)
The archetypical simple example is FinDimVect itself, the category of finite dimensional vector spaces over some ground field $k$. This has a single isomorphism class of simple objects: given by $k$ itself.
The category of finite-dimensional complex representations of a compact Lie group $G$ is semisimple, with the simple objects being precisely the irreducible representations (this is the content of Schur's lemma). If $G$ is not a compact Lie group, one needs to pass from the concept of ‘direct sum’ to ‘direct integral’.
Every fusion category is a semisimple category.
Various alternative definitions of semisimple category appear in the literature, and a number are compared here:
For example, he compares a definition of “abelian semisimple category”(an abelian category where every object is a direct sum of a possibly infinite number of simple objects) with “Müger semisimple” categories as discussed here:
There is a related discussion on the nForum and a discussion on MathOverflow.
Last revised on June 18, 2023 at 18:01:07. See the history of this page for a list of all contributions to it.