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Comodule

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In mathematics, a comodule or corepresentation is a concept dual to a module. The definition of a comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra.

Contents

Formal definition

Let K be a field, and C be a coalgebra over K. A (right) comodule over C is a K-vector space M together with a linear map

ρ : M → M ⊗ C

such that

  1. ( i d ⊗ Δ ) ∘ ρ = ( ρ ⊗ i d ) ∘ ρ
  2. ( i d ⊗ ε ) ∘ ρ = i d ,

where Δ is the comultiplication for C, and ε is the counit.

Note that in the second rule we have identified M ⊗ K with M .

Examples

  • A coalgebra is a comodule over itself.
  • If M is a finite-dimensional module over a finite-dimensional K-algebra A, then the set of linear functions from A to K forms a coalgebra, and the set of linear functions from M to K forms a comodule over that coalgebra.
  • A graded vector space V can be made into a comodule. Let I be the index set for the graded vector space, and let C I be the vector space with basis e i for i ∈ I . We turn C I into a coalgebra and V into a C I -comodule, as follows:
    1. Let the comultiplication on C I be given by Δ ( e i ) = e i ⊗ e i .
    2. Let the counit on C I be given by ε ( e i ) = 1   .
    3. Let the map ρ on V be given by ρ ( v ) = ∑ v i ⊗ e i , where v i is the i-th homogeneous piece of v .

    Rational comodule

    If M is a (right) comodule over the coalgebra C, then M is a (left) module over the dual algebra C∗, but the converse is not true in general: a module over C∗ is not necessarily a comodule over C. A rational comodule is a module over C∗ which becomes a comodule over C in the natural way.

    References

    Comodule Wikipedia
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