Coalgebra - Knowlpedia

Definition

Let $k$ be a commutative ring. A $k$ -coalgebra is a $k$ -module $C$ equipped with $k$ -linear maps

\Delta:C\longrightarrow C\otimes_k C, \qquad \varepsilon:C\longrightarrow k,

called the comultiplication and counit, such that

(\Delta\otimes\operatorname{id})\Delta = (\operatorname{id}\otimes\Delta)\Delta

and

(\varepsilon\otimes\operatorname{id})\Delta =\operatorname{id}_C = (\operatorname{id}\otimes\varepsilon)\Delta,

after the canonical identifications $k\otimes_k C\cong C\cong C\otimes_k k$ . Thus a coalgebra is a comonoid in the monoidal category of $k$ -modules.

How to read the axioms

If one writes $\Delta(c)=\sum c_{(1)}\otimes c_{(2)}$ in Sweedler notation, coassociativity says that iterating $\Delta$ has an unambiguous value $\sum c_{(1)}\otimes c_{(2)}\otimes c_{(3)}$ . The counit equations say

\sum\varepsilon(c_{(1)})c_{(2)}=c =\sum c_{(1)}\varepsilon(c_{(2)}).

These formulas are notation for identities of maps; they do not require a preferred finite expansion of $\Delta(c)$ .

Morphisms and cocommutativity

A coalgebra homomorphism $f:C\to D$ is a $k$ -linear map satisfying

\Delta_Df=(f\otimes f)\Delta_C, \qquad \varepsilon_Df=\varepsilon_C.

The coalgebra is cocommutative when $\tau\Delta=\Delta$ , where $\tau(x\otimes y)=y\otimes x$ .

Duality with algebras

When $C$ is finitely generated projective over $k$ , dualizing $\Delta$ and $\varepsilon$ makes $C^\vee=\operatorname{Hom}_k(C,k)$ an associative unital algebra. Without finiteness or projectivity, the natural comparison $C^\vee\otimes C^\vee\to(C\otimes C)^\vee$ need not be an isomorphism, so naive linear duality does not exchange arbitrary coalgebras and algebras. Topological or finite-dual constructions are then needed.

Examples

Every set $X$ gives a coalgebra with basis $\{e_x:x\in X\}$ , $\Delta(e_x)=e_x\otimes e_x$ , and $\varepsilon(e_x)=1$ .
If $A$ is a finite-dimensional $k$ -algebra over a field, the dual vector space $A^\vee$ is a coalgebra by transposing multiplication and the unit.

References

Moss E. Sweedler, Hopf Algebras, W. A. Benjamin, 1969. Relevant: Chapter 1 on coalgebras and their dual algebras.
Christian Kassel, Quantum Groups, Graduate Texts in Mathematics 155, Springer, 1995. Publisher record. Relevant: Chapter III, §1.