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

$\rho\colon M \to M \otimes C$

such that

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

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

Note that in the second rule we have identified $M \otimes K$ 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 $\Delta(e_i) = e_i \otimes e_i$.
2. Let the counit on $C_{I}$ be given by $\varepsilon(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 $v$.

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.