Comodule


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

Let K be a field, and C be a coalgebra over K. A comodule over C is a K-vector space M together with a linear map
such that
  2. ,
where Δ is the comultiplication for C, and ε is the counit.
Note that in the second rule we have identified with.

Examples

If M is a comodule over the coalgebra C, then M is a 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.