nLab
semisimple category

Context

Category theory

Additive and abelian categories

Monoidal categories

Contents

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

An abelian category is called semisimple if every object is a semisimple object, hence a direct sum of finitely many simple objects. See semisimple abelian category.

Alternatively, a monoidal linear category (that is, a monoidal category enriched over Vect) is called semisimple if:

  • it has finite biproducts (usually called ‘direct sums’),

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

  • there exist objects X iX_i labeled by an index set II such that Hom(X i,X j)δ ijkHom(X_i, X_j) \cong \delta_{ij} k where kk is the ground field (such objects are called simple) and such that for any two objects VV and WW in the category, the natural composition map

    (1) iIHom(V,X i)Hom(X i,W)Hom(V,W) \bigoplus_{i \in I} Hom(V, X_i) \otimes Hom(X_i, W) \rightarrow Hom(V, W)

    is an isomorphism.

Direct sums of simple objects

Note that this definition implies that every object VV is a direct sum of simple objects X iX_i. To see this, note that the third item of the definition is equivalent to stipulating that the vector space Hom(X i,V)Hom(X_i, V) is in canonical duality with the vector space Hom(V,X i)Hom(V, X_i). Indeed, we have a canonical pairing

(2)Hom(V,X i)Hom(X i,V)k Hom(V, X_i) \otimes Hom(X_i, V) \rightarrow k

given by sending fgfgf \otimes g \mapsto \langle f \circ g \rangle where the ”\langle \cdot \rangle” notation refers to extracting scalars from endomorphisms of simple objects (each such endomorphism is a scalar multiple of the identity). We also have a canonical copairing

(3)kHom(X i,V)Hom(V,X i) k \rightarrow Hom(X_i, V) \otimes Hom(V, X_i)

given by sending id X i\id_{X_i} to the ”iith block” of the image of the identity id V\id_V arrow under the isomorphism given in the definition. One can check that this pairing and copairing satisfy the snake equations. Hence if we choose a basis

(4){a i,p:X iV} \{ a_{i,p} : X_i \rightarrow V \}

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

(5){a i p:VX i} \{ a_i^p : V \rightarrow X_i \}

satisfying

(6)a i pa j,q=δ ijδ p qand i,pa i,pa i p=id V. a_i^p a_{j,q} = \delta_{ij} \delta_p^q \quad and \quad \sum_{i,p} a_{i,p} a_i^p = \id_V.

This says precisely that VV has been expressed as a direct sum of the X iX_i.

Remarks

  • The above definition definition of semisimple monoidal linear category (taken from the reference of Müger below) 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)Hom(V, W) must at least have the same dimension asHom(W,V)Hom(W,V). This is the easiest way to check if a category will fail to be semisimple. For instance, the category Rep(A)Rep(A) of representations of an algebra AA will rarely be semisimple, precisely because there is no relation between Hom(V,W)Hom(V, W) and Hom(W,V)Hom(W,V) in general. Again, this can be traced back to the original algebra AA 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:CCS: C \rightarrow C from the category to itself with the property that there are canonical isomorphisms

    (7)Hom(V,W)Hom(W,SV) Hom(V, W) \cong Hom(W, SV)^\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.

  • For 2-Hilbert spaces, there is an antilinear **-operation on the hom-sets *:Hom(V,W)Hom(W,V)* : 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

References

There is related discussion on the nnForum here.

Revised on October 29, 2013 22:31:40 by Urs Schreiber (82.169.114.243)