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
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
- ,
where Δ is the comultiplication for C, and ε is the counit.
Note that in the second rule we have identified with .
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 be the vector space with basis for . We turn into a coalgebra and V into a -comodule, as follows:
-
- Let the comultiplication on be given by .
- Let the counit on be given by .
- Let the map on V be given by , where is the i-th homogeneous piece of .
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
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.