Elongated pentagonal gyrocupolarotunda

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

$\rho :M\to M\otimes C$

such that

1. $(id\otimes \mathrm{\Delta })\circ \rho =(\rho \otimes id)\circ \rho$
2. $(id\otimes \epsilon )\circ \rho =id$,

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

Note that in the second rule we have identified $M\otimes K$ 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 $i\in I$. 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 $\mathrm{\Delta }(e_i)=e_i\otimes e_i$.
2. Let the counit on $C_I$ be given by $\epsilon (e_i)=1$.
3. Let the map $\rho$ on V be given by $\rho (v)=\sum v_i\otimes e_i$, where $v_i$ is the i-th homogeneous piece of $v$.

References

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