nLab semisimple category

Idea

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.

Definition

Definition

(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 $k$:

Definition

(semisimple linear category)
A linear category (that is, a category enriched over Vect, cf. tensor category) is called semisimple if:

1. it has finite biproducts

   (usually called ‘direct sums’),

2. idempotents split

   (so we say that it ‘has subobjects’ or, perhaps better, ‘has retracts’),

3. there exist objects $X_i$ labeled by an index set $I$ such that

   1. for any pair $i, j$ of indices we have an isomorphism

      (1)$$Hom(X_i, X_j) \;\cong\; \delta_{i j} k \,,$$

      where $\delta$ denotes the Kronecker delta and $k$ the ground field

      (such objects are called simple),

   2. for any pair of objects $V,\,W$, the natural composition map

      (2)$$\bigoplus_{i \in I} \, Hom(V, X_i) \,\otimes\, Hom(X_i, W) \longrightarrow Hom(V, W)$$

      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))

Proposition

(direct sum decomposition into simple objects)
Definition implies that every object $V$ is a direct sum of simple objects $X_i$.

Proof

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:

1. a pairing

   $$\begin{array}{ccc} Hom(V, X_i) \,\otimes\, Hom(X_i, V) & \longrightarrow & Hom(X_i, X_i) &\simeq& k \\ f \,\otimes\, g &\mapsto& f \,\circ\, g \mathrlap{\,,} \end{array}$$

   where on the right we used (1),

2. a copairing

   $$k \cdot Id_B \;\hookrightarrow\; Hom(V,V) \longrightarrow Hom(X_i, V) \otimes Hom(V, X_i) \mathrlap{\,,}$$

   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

$$\big\{ a_{i,p} \,\colon\, X_i \rightarrow V \big\}$$

for each vector space $Hom(X_i, V)$, we get a corresponding dual basis

$$\big\{ a_i^p \,\colon\, V \rightarrow X_i \big\}$$

satisfying

$$a_i^p a_{j,q} \;=\; \delta_{i j} \delta_p^q \;\;\; \text{and} \;\;\; \sum_{i,p} a_{i,p} a_i^p \;=\; \id_V \mathrlap{\,.}$$

This says precisely that $V$ has been expressed as a direct sum of the $X_i$.

Remark

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.

Remark

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.

Remark

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

$$Hom(V, W) \cong Hom(W, S V)^\vee$$

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.

Remark

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)

Examples

• 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.

References

Various alternative definitions of semisimple category appear in the literature, and a number are compared here:

• James Bailie, Semisimple categories (pdf.

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:

• Michael Müger, p. 6 in: From Subfactors to Categories and Topology I. Frobenius algebras in and Morita equivalence classes of tensor categories, J. Pure Appl. Alg. 180 (2003) 81-157 [arXiv:math/0111204, doi:10.1016/S0022-4049(02)00247-5]

There is a related discussion on the nForum and a discussion on MathOverflow.