Coalgebra

Coalgebras occur naturally in a number of contexts (for example, Group Scheme s).

There are also F-coalgebra s, with important applications in Computer Science .

FORMAL DEFINITION

Formally, a coalgebra over a Field ''K'' is a ''K''- Vector Space ''C'' together with ''K''- Linear Map s

\Delta : C 	o C \otimes_K C

and

\epsilon : C 	o K

such that

#

(\mathrm{id}_C \otimes \Delta) \circ \Delta = (\Delta \otimes \mathrm{id}_C) \circ \Delta

(\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

C\otimes (C\otimes C)

with

(C \otimes C)\otimes C

; the two are naturally isomorphic. Similarly, in the second diagram the naturally isomorphic spaces

C

C\otimes K

and

K\otimes C

are identified.

The first diagram is the dual of the one expressing Associativity of algebra multiplication (called the coassociativity of the comultiplication); the second diagram is the dual of the one expressing the existence of a multiplicative Identity . Accordingly, the map Δ is called the comultiplication (or '''coproduct''') 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
:

\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 (checking this is a good way to get used to the axioms).

As a second example, consider the Polynomial Ring ''K'' {Link without Title} in one Indeterminate ''X''. This becomes a coalgebra if we define

:

\Delta(X^n) = \sum_{m=0}^n X^m\otimes X^{n-m}

and
:

\epsilon(X^n)=\begin{cases}1& \mbox{if } n=0\
0& \mbox{if } n>0
\end{cases}

for all

n\ge 0.

Again, because of linearity, this suffices to define Δ and ε uniquely on all of ''K'' Now ''K''[''X'' is both a unital associative algebra and a coalgebra, and the two structures are compatible. Objects like this are called Bialgebra s, and in fact most of the important coalgebras considered in practice are bialgebras. Examples include Hopf Algebra s and Lie Bialgebra s.

consisting of all ''K''-linear maps from ''A'' to ''K'' is a coalgebra. The multiplication of ''A'' can be viewed as a linear map $A\otimes A
ightarrow A$ , which when dualized yields a linear map $A^---
ightarrow (A\otimes A)^---$ . In the finite-dimensional case, $(A\otimes A)^---$ is naturally isomorphic to $A^---\otimes A^---$ , so we have defined a comultiplication on ''A''^---. The counit of ''A''^--- is given by evaluating Linear Function als 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
:

\Delta(c)=\sum_i c_{(1)}^{(i)}\otimes c_{(2)}^{(i)}.

In Sweedler's notation, this is abbreviated to
:

\Delta(c)=\sum_{(c)} c_{(1)}\otimes c_{(2)}.

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

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

The coassociativity of Δ can be expressed as
:

\sum_{(c)}c_{(1)}\otimes\left(\sum_{(c_{(2)})}(c_{(2)})_{(1)}\otimes (c_{(2)})_{(2)}
ight) = \sum_{(c)}\left( \sum_{(c_{(1)})}(c_{(1)})_{(1)}\otimes (c_{(1)})_{(2)}
ight) \otimes c_{(2)}.

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

\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
:

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

and
:

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 : ''C''⊗''C'' → ''C''⊗''C'' is the ''K''-linear map defined by σ(''c''⊗''d'') = ''d''⊗''c'' for all ''c'',''d'' in ''C''. In Sweedler's sumless notation, ''C'' is co-commutative if and only if
:

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 (''C''₁,Δ₁,ε₁) and (''C''₂,Δ₂,ε₂) are two coalgebras over the same field ''K'', then a coalgebra morphism from ''C''₁ to ''C''₂ is a ''K''-linear map ''f'' : ''C''₁ → ''C''₂ such that (''f'' ⊗ ''f'') o Δ₁ = Δ₂ o ''f'' and ε₂ o ''f'' = ε₁.
In Sweedler's sumless notation, the first of these properties may be written as:
:

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'')⊆''I''⊗''C'' + ''C''⊗''I''. 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'')⊆''D''⊗''D''; in that case, ''D'' is itself a coalgebra, with the restriction of ε to ''D'' as counit.

The is a subcoalgebra of ''C''₂. The common Isomorphism Theorem s are valid for coalgebras, so for instance ''C''₁/ker(''f'') is isomorphic to im(''f'').

As we have seen above, if ''A'' is a finite-dimensional unital associative ''K''-algebra, then

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.

Corresponding to the concept of Representation for algebras is a corepresentation of a coalgebra. An ''n''-dimensional corepresentation of a coalgebra ''C'' is an ''n''-by-''n'' matrix ''u'' with elements in ''C'' (i.e.

u \in M_n(C)

) such that

$\Delta(u_{ij}) = \sum_{k=1}^n u_{ik} \otimes u_{kj}$ for $i, j = 1,\dots,n$ ,

$\epsilon(u_{ij}) = \delta_{oj}$ for $i, j = 1,\dots,n$ , where $\delta_{ij}$ is the Kronecker Delta .

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Coalgebra