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 (right) comodule over C is a K-vector space M together with a linear map

${\displaystyle \rho \colon M\to M\otimes C}$

such that

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

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

Note that in the second rule we have identified ${\displaystyle M\otimes K}$ with ${\displaystyle M\,}$.

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 ${\displaystyle C_{I}}$ be the vector space with basis ${\displaystyle e_{i}}$ for ${\displaystyle i\in I}$. We turn ${\displaystyle C_{I}}$ into a coalgebra and V into a ${\displaystyle C_{I}}$-comodule, as follows:

1. Let the comultiplication on ${\displaystyle C_{I}}$ be given by ${\displaystyle \Delta (e_{i})=e_{i}\otimes e_{i}}$.
2. Let the counit on ${\displaystyle C_{I}}$ be given by ${\displaystyle \varepsilon (e_{i})=1\ }$.
3. Let the map ${\displaystyle \rho }$ on V be given by ${\displaystyle \rho (v)=\sum v_{i}\otimes e_{i}}$, where ${\displaystyle v_{i}}$ is the i-th homogeneous piece of ${\displaystyle v}$.

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

• Gómez-Torrecillas, José (1998), "Coalgebras and comodules over a commutative ring", Revue Roumaine de Mathématiques Pures et Appliquées, 43: 591–603
• Montgomery, Susan (1993). Hopf algebras and their actions on rings. Regional Conference Series in Mathematics. 82. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029.
• Sweedler, Moss (1969), Hopf Algebras, New York: W.A.Benjamin