Coalgebra

In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras.

Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions (see below).

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

There are also F-coalgebras, with important applications in computer science.

Formal definition

Formally, a coalgebra over a field K is a vector space C over K together with K-linear maps Δ: C → C ⊗ C and ε: C → K such that

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

(Here ⊗ refers to the tensor product over K and id is the identity function.)

Equivalently, the following two diagrams commute:

In the first diagram we silently identify C ⊗ (C ⊗ C) with (C ⊗ C) ⊗ C; the two are naturally isomorphic.^[1] Similarly, in the second diagram the naturally isomorphic spaces C, C ⊗ K and K ⊗ C are identified.^[2]

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 = K^(S). We define

Δ(s) = s ⊗ s and ε(s) = 1 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[X] in one indeterminate X. This becomes a coalgebra (the divided power coalgebra^[3]^[4]) if for all n ≥ 0 we define:

\Delta(X^n) = \sum_{k=0}^n \dbinom{n}{k} X^k\otimes X^{n-k},

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

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. Examples include Hopf algebras and Lie bialgebras.

The tensor algebra and the exterior algebra are further examples of coalgebras.

In some cases the singular homology of a topological space forms a coalgebra.^[5]

If C is the K-vector space with basis {s, c}, consider Δ: C → C ⊗ C is given by

Δ(s) = s ⊗ c + c ⊗ s

Δ(c) = c ⊗ c − s ⊗ s

and ε: C → K is given by

ε(s) = 0

ε(c) = 1

In this situation, (C, Δ, ε) is a coalgebra known as trigonometric coalgebra.^[6]^[7]

For a locally finite poset P with set of intervals J we can define the incidence coalgebra C with J as basis and comultiplication for x < z

\Delta[x,z] = \sum_{x < y < z} [x,y] \otimes [y,z] \ .

The intervals of length zero correspond to points of P and are group-like elements.^[8]

Finite dimensions

In finite dimensions, the duality between algebras and coalgebras is closer: the dual of a finite-dimensional (unital associative) algebra is a coalgebra, while the dual of a finite-dimensional coalgebra is a (unital associative) algebra. In general, the dual of an algebra may not be a coalgebra.

The key point is that in finite dimensions, (A ⊗ A)* and A* ⊗ A* are isomorphic.

To distinguish these: in general, algebra and coalgebra are dual notions (meaning that their axioms are dual: reverse the arrows), while for finite dimensions, they are dual objects (meaning that a coalgebra is the dual object of an algebra and conversely).

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 A ⊗ A → A, which when dualized yields a linear map A* → (A ⊗ A)*. In the finite-dimensional case, (A ⊗ A)* is naturally isomorphic to A* ⊗ 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

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

In Sweedler's notation,^[9] 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)} \varepsilon(c_{(1)})c_{(2)} = \sum_{(c)} c_{(1)}\varepsilon(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)}\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

\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=\varepsilon(c_{(1)})c_{(2)} = c_{(1)}\varepsilon(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 $\sigma\circ\Delta = \Delta$ , 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.)

A group-like element is an element x such that Δ(x) = x⊗x and ε(x) = 1. A primitive element x satisfies Δ(x) = x⊗1 + 1⊗x.^[10]^[11]

If (C₁, Δ₁, ε₁) and (C₂, Δ₂, ε₂) are two coalgebras over the same field K, then a coalgebra morphism from C₁ to C₂ is a K-linear map f : C₁ → C₂ such that $(f\otimes f)\circ\Delta_1 = \Delta_2\circ f$ and $\epsilon_2\circ f = \epsilon_1$ . 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 linear 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 kernel of every coalgebra morphism f : C₁ → C₂ 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).

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 subcoalgebras, 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 or comodule.

References

↑ Yokonuma (1992). Prop. 1.7. p. 12.
↑ Yokonuma (1992). Prop. 1.4. p. 10.
↑ See also Dăscălescu, Năstăsescu & Raianu (2001). Hopf Algebras: An introduction. p. 3.
↑ See also Raianu, Serban. Coalgebras from Formulas, p. 2.
↑ Lecture notes for reference
↑ See also Dăscălescu, Năstăsescu & Raianu (2001). Hopf Algebras: An introduction. p. 4. , and Dăscălescu, Năstăsescu & Raianu (2001). Hopf Algebras: An introduction. p. 55. , Ex. 1.1.5.
↑ Raianu, Serban. Coalgebras from Formulas, p. 1.
↑ Montgomery (1993) p.61
↑ Underwood (2011) p.35
↑ Mikhalev, Aleksandr Vasilʹevich; Pilz, Günter, eds. (2002). The Concise Handbook of Algebra. Springer-Verlag. p. 307, C.42. ISBN 0792370724.
↑ Abe, Eiichi (2004). Hopf Algebras. Cambridge Tracts in Mathematics 74. Cambridge University Press. p. 59. ISBN 0-521-60489-3.

External links

William Chin: A brief introduction to coalgebra representation theory