# Coalgebra

In mathematics, coalgebras are structures that are in a certain sense dual to the unital associative algebras. One formulates the axioms of unital associative algebras in terms of commutative diagrams, and then turns all arrows around to get the axioms of coalgebras.

Coalgebras occur naturally in a number of contexts (for example, group schemes).

There are also F-Coalgebras.

## Formal definition

Formally, a coalgebra over a field K is a K-vector space C together with K-linear maps ${\displaystyle \Delta :C\to C\otimes _{K}C}$ and ${\displaystyle \epsilon :C\to K}$ such that

1. ${\displaystyle (\mathrm {id} _{C}\otimes \Delta )\circ \Delta =(\Delta \otimes \mathrm {id} _{C})\circ \Delta }$
2. ${\displaystyle (\mathrm {id} _{C}\otimes \epsilon )\circ \Delta =\mathrm {id} _{C}=(\epsilon \otimes \mathrm {id} _{C})\circ \Delta }$.

Equivalently, the following two diagrams commute:

In the first diagram we silently identify ${\displaystyle C\otimes (C\otimes C)}$ with ${\displaystyle (C\otimes C)\otimes C}$; the two are naturally isomorphic. Similarly, in the second diagram the naturally isomorphic spaces ${\displaystyle C}$, ${\displaystyle C\otimes K}$ and ${\displaystyle K\otimes C}$ are identified.

The first diagram is the dual of the one expressing associativity of algebra multiplication; the second diagram is the dual of the one expressing the existence of a multiplicative identity. Accordingly, the map Δ is called the comultiplication of C and ε is the counit of C.

## Examples

Take an arbitrary set S and form the K-vector space with basis S. The elements of this vector space are those functions from S to K that map all but finitely many elements of S to zero; we identify the element s of S with the function that maps s to 1 and all other elements of S to 0. We will denote this space by C. We define

${\displaystyle \Delta (s)=s\otimes s\quad {\mbox{ and }}\quad \epsilon (s)=1\quad {\mbox{ for all }}s\in S.}$

By linearity, both Δ and ε can then uniquely be extended to all of C. The vector space C becomes a coalgebra with comultiplication Δ and counit ε (you may want to check this to get used to the axioms).

As a second example, consider the polynomial ring K[X] in one indeterminate X. This becomes a coalgebra if we define

${\displaystyle \Delta (X^{n})=\sum _{m=0}^{n}X^{m}\otimes X^{n-m}\quad {\mbox{ and }}\quad \epsilon (X^{n})={\begin{cases}1&{\mbox{if }}n=0\\0&{\mbox{if }}n>0\end{cases}}\quad {\mbox{ for all }}n\geq 0.}$

Again, because of linearity, this suffices to define Δ and ε uniquely on all of K[X]. Now K[X] is both a unital associative algebra and a coalgebra, and the two structures are compatible. Objects like this are called bialgebras, and in fact most of the important coalgebras considered in practice are bialgebras. For more examples of coalgebras, see the articles on bialgebras and on Hopf algebras (which are special bialgebras).

If A is a finite-dimensional unital associative K-algebra, then its K-dual A* consisting of all K-linear maps from A to K is a coalgebra. The multiplication of A can be viewed as a linear map ${\displaystyle A\otimes A\rightarrow A}$, which when dualized yields a linear map ${\displaystyle A^{*}\rightarrow (A\otimes A)^{*}}$. In the finite-dimensional case, ${\displaystyle (A\otimes A)^{*}}$ is naturally isomorphic to ${\displaystyle A^{*}\otimes A^{*}}$, so we have defined a comultiplication on A*. The counit of A* is given by evaluating linear functionals at 1.

## Sweedler notation

When working with coalgebras, a certain notation for the comultiplication simplifies the formulas considerably and has become quite popular. Given an element c of the coalgebra (C,Δ,ε), we know that there exist elements c(1)(i) and c(2)(i) in C such that

${\displaystyle \Delta (c)=\sum _{i}c_{(1)}^{(i)}\otimes c_{(2)}^{(i)}.}$

In Sweedler's notation, this is abbreviated to

${\displaystyle \Delta (c)=\sum _{(c)}c_{(1)}\otimes c_{(2)}.}$

The fact that ε is a counit can then be expressed with the following formula

${\displaystyle c=\sum _{(c)}\epsilon (c_{(1)})c_{(2)}=\sum _{(c)}c_{(1)}\epsilon (c_{(2)}).\;}$

The coassociativity of Δ can be expressed as

${\displaystyle \sum _{(c)}c_{(1)}\otimes \left(\sum _{(c_{(2)})}(c_{(2)})_{(1)}\otimes (c_{(2)})_{(2)}\right)=\sum _{(c)}\left(\sum _{(c_{(1)})}(c_{(1)})_{(1)}\otimes (c_{(1)})_{(2)}\right)\otimes c_{(2)}.}$

In Sweedler's notation, both of these expressions are written as

${\displaystyle \sum _{(c)}c_{(1)}\otimes c_{(2)}\otimes c_{(3)}.}$

Some authors omit the summation symbols as well; in this sumless Sweedler notation, we may write

${\displaystyle \Delta (c)=c_{(1)}\otimes c_{(2)}}$

and

${\displaystyle c=\epsilon (c_{(1)})c_{(2)}=c_{(1)}\epsilon (c_{(2)}).\;}$

Whenever a variable with lowered and parenthesized index is encountered in an expression of this kind, a summation symbol for that variable is implied.

## Further concepts and facts

A coalgebra (C,Δ,ε) is called co-commutative if σoΔ = Δ, where σ : CCCC is the K-linear map defined by σ(cd) = dc for all c,d in C. In Sweedler's sumless notation, C is co-commutative if and only if

${\displaystyle c_{(1)}\otimes c_{(2)}=c_{(2)}\otimes c_{(1)}\;}$

for all c in C. (It's important to understand that the implied summation is significant here: we are not requiring that all the summands are pairwise equal, only that the sums are equal, a much weaker requirement.)

If (C111) and (C222) are two coalgebras over the same field K, then a coalgebra morphism from C1 to C2 is a K-linear map f : C1C2 such that (ff) o Δ1 = Δ2 o f and ε2 o f = ε1. In Sweedler's sumless notation, the first of these properties may be written as:

${\displaystyle f(c_{(1)})\otimes f(c_{(2)})=f(c)_{(1)}\otimes f(c)_{(2)}.}$

The composition of two coalgebra morphisms is again a coalgebra morphism, and the coalgebras over K together with this notion of morphism form a category.

A subspace I in C is called a coideal if I⊆ker(ε) and Δ(I)⊆IC + CI. In that case, the quotient space C/I becomes a coalgebra in a natural fashion.

A subspace D of C is called a subcoalgebra if Δ(D)⊆DD; in that case, D is itself a coalgebra, with the restriction of ε to D as counit.

The kernel of every coalgebra morphism f : C1C2 is a coideal in C1, and the image is a subcoalgebra of C2. The common isomorphism theorems are valid for coalgebras, so for instance C1/ker(f) is isomorphic to im(f).

As we have seen above, if A is a finite-dimensional unital associative K-algebra, then A* is a finite-dimensional coalgebra, and indeed every finite-dimensional coalgebra arises in this fashion from some finite-dimensional algebra (namely from the coalgebra's K-dual). Under this correspondence, the commutative finite-dimensional algebras correspond to the cocommutative finite-dimensional coalgebras. So in the finite-dimensional case, the theories of algebras and of coalgebras are dual; studying one is equivalent to studying the other. However, things diverge in the infinite-dimensional case: while the K-dual of every coalgebra is an algebra, the K-dual of an infinite-dimensional algebra need not be a coalgebra.

Every coalgebra is the sum of its finite-dimensional coalgebras, something that's not true for algebras. In a certain sense then, coalgebras are generalizations of (duals of) finite-dimensional unital associative algebras.