semisimple category in nLab (original) (raw)

Contents

Context

Category theory

category theory

Additive and abelian 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

Alternatively, at least over an algebraically closed ground field kk:

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 iX_i labeled by an index set II such that
    1. for any pair i,ji, j of indices we have an isomorphism
      (1)Hom(X i,X j)≅δ ijk, Hom(X_i, X_j) \;\cong\; \delta_{i j} k \,,
      where δ\delta denotes the Kronecker delta and kk the ground field
      (such objects are called simple),
    2. for any pair of objects V,WV,\,W, the natural composition map
      (2)⨁ i∈IHom(V,X i)⊗Hom(X i,W)⟶Hom(V,W) \bigoplus_{i \in I} \, Hom(V, X_i) \,\otimes\, Hom(X_i, W) \longrightarrow Hom(V, W)
      is an isomorphism.
      (If the field kk is not algebraically closed, a simple object XX can have a finite-dimensional division algebra other than kk itself as its endomorphism algebra.)

(e.g. Müger (2003, p. 6))

Proposition

(direct sum decomposition into simple objects)
Definition implies that every object VV is a direct sum of simple objects X iX_i.

Proof

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:

  1. a pairing
    Hom(V,X i)⊗Hom(X i,V) ⟶ Hom(X i,X i) ≃ k f⊗g ↦ f∘g, \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⋅Id B↪Hom(V,V)⟶Hom(X i,V)⊗Hom(V,X i), 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

{a i,p:X i→V} \big\{ a_{i,p} \,\colon\, X_i \rightarrow V \big\}

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

{a i p:V→X i} \big\{ a_i^p \,\colon\, V \rightarrow X_i \big\}

satisfying

a i pa j,q=δ ijδ p qand∑ i,pa i,pa i p=id V. 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 VV has been expressed as a direct sum of the X iX_i.

Examples

References

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.