 Coalgebra

In mathematics, coalgebras or cogebras are structures that are dual (in the 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 Fcoalgebras, with important applications in computer science.
Contents
Formal definition
Formally, a coalgebra over a field K is a vector space C over K together with Klinear maps and such that
 .
(Here and refer 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 with ; the two are naturally isomorphic.^{[1]} Similarly, in the second diagram the naturally isomorphic spaces C , and 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 Kvector 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
 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 we define
 In some cases the singular homology of a topological space forms a coalgebra.^{[5]}
 If C is the Kvector space with basis {s, c}, consider is given by
Finite dimensions
In finite dimensions, the duality between algebras and coalgebras is closer: the dual of a finitedimensional (unital associative) algebra is a coalgebra, while the dual of a finitedimensional 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, .
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 finitedimensional unital associative Kalgebra, then its Kdual A^{*} consisting of all Klinear maps from A to K is a coalgebra. The multiplication of A can be viewed as a linear map , which when dualized yields a linear map . In the finitedimensional case, is naturally isomorphic to , 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
In Sweedler's notation, this is abbreviated to
The fact that ε is a counit can then be expressed with the following formula
The coassociativity of Δ can be expressed as
In Sweedler's notation, both of these expressions are written as
Some authors omit the summation symbols as well; in this sumless Sweedler notation, we may write
and
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 cocommutative if , where is the Klinear map defined by for all c,d in C. In Sweedler's sumless notation, C is cocommutative if and only if
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 and are two coalgebras over the same field K, then a coalgebra morphism from C_{1} to C_{2} is a Klinear map such that and . In Sweedler's sumless notation, the first of these properties may be written as:
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 : C_{1} → C_{2} is a coideal in C_{1}, and the image is a subcoalgebra of C_{2}. The common isomorphism theorems are valid for coalgebras, so for instance C_{1}/ker(f) is isomorphic to im(f).
If A is a finitedimensional unital associative Kalgebra, then A^{*} is a finitedimensional coalgebra, and indeed every finitedimensional coalgebra arises in this fashion from some finitedimensional algebra (namely from the coalgebra's Kdual). Under this correspondence, the commutative finitedimensional algebras correspond to the cocommutative finitedimensional coalgebras. So in the finitedimensional case, the theories of algebras and of coalgebras are dual; studying one is equivalent to studying the other. However, things diverge in the infinitedimensional case: while the Kdual of every coalgebra is an algebra, the Kdual of an infinitedimensional algebra need not be a coalgebra.
Every coalgebra is the sum of its finitedimensional coalgebras, something that's not true for algebras. In a certain sense then, coalgebras are generalizations of (duals of) finitedimensional unital associative algebras.
Corresponding to the concept of representation for algebras is a corepresentation or comodule.
Notes
 ^ Yokonuma (1992), p. 12, Prop. 1.7.
 ^ Yokonuma (1992), p. 10, Prop. 1.4.
 ^ See also Dăscălescu, Năstăsescu & Raianu (2001), p. 3.
 ^ See also Raianu, Serban. Coalgebras from Formulas, p. 2.
 ^ Lecture notes for reference
 ^ See also Dăscălescu, Năstăsescu & Raianu (2001), p. 4 & p. 55, Ex. 1.1.5.
 ^ Raianu, Serban. Coalgebras from Formulas, p. 1.
See also
References
 Block, Richard E.; Leroux, Pierre (1985), "Generalized dual coalgebras of algebras, with applications to cofree coalgebras", Journal of Pure and Applied Algebra 36 (1): 15–21, doi:10.1016/00224049(85)90060X, ISSN 00224049, MR782637, http://dx.doi.org/10.1016/00224049(85)90060X
 Dăscălescu, Sorin; Năstăsescu, Constantin; Raianu, Șerban (2001), Hopf Algebras, Pure and Applied Mathematics, 235 (1st ed.), Marcel Dekker, ISBN 0824704819.
 Hazewinkel, Michiel (2003), "Cofree coalgebras and multivariable recursiveness", Journal of Pure and Applied Algebra 183 (1): 61–103, doi:10.1016/S00224049(03)000136, ISSN 00224049, MR1992043, http://dx.doi.org/10.1016/S00224049(03)000136
 Yokonuma, Takeo (1992), Tensor spaces and exterior algebra, Translations of mathematical monographs, 108, AMS Bookstore, ISBN 9780821845646.
 Chapter III, section 11 in Bourbaki, Nicolas (1989). Algebra. SpringerVerlag. ISBN 0387193731.
External links
 William Chin: A brief introduction to coalgebra representation theory
Categories: Coalgebras
Wikimedia Foundation. 2010.