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.

Formal definition[edit]

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

such that

  1. ,

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

Note that in the second rule we have identified with .

Examples[edit]

  • 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 be the vector space with basis for . We turn into a coalgebra and V into a -comodule, as follows:
  1. Let the comultiplication on be given by .
  2. Let the counit on be given by .
  3. Let the map on V be given by , where is the i-th homogeneous piece of .

Rational comodule[edit]

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[edit]