Coalgebra

In mathematics, coalgebras or co-gebras 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.

Every coalgebra, by (vector space) duality, give rise to an algebra, but not in general the other way.

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

A primary task, of practical use in physics, is to obtain combinations of systems with different states of angular momentum and spin.

This is provided by the total angular momentum operator, which extracts the needed quantity from each side of the tensor product.

Examining the lifting in detail, one observes that the coproduct behaves as the shuffle product, essentially because the two factors above, the left and right

must be kept in sequential order during products of multiple angular momenta (rotations are not commutative).

The formal definition of the coalgebra, below, abstracts away this particular special case, and its requisite properties, into a general setting.

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

[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. Take an arbitrary set S and form the K-vector space C = K(S) with basis S, as follows.

Now K[X] is both a unital associative algebra and a coalgebra, and the two structures are compatible.

The singular homology of a topological space forms a graded coalgebra whenever the Künneth isomorphism holds, e.g. if the coefficients are taken to be a field.

[6][7] For a locally finite poset P with set of intervals J, define the incidence coalgebra C with J as basis.

The comultiplication and counit are defined as The intervals of length zero correspond to points of P and are group-like elements.

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.

In the finite-dimensional case, (A ⊗ A)∗ is naturally isomorphic to A∗ ⊗ A∗, so this defines a comultiplication on A∗.

When working with coalgebras, a certain notation for the comultiplication simplifies the formulas considerably and has become quite popular.

In Sweedler's notation,[9] (so named after Moss Sweedler), this is abbreviated to The fact that ε is a counit can then be expressed with the following formula Here it is understood that the sums have the same number of terms, and the same lists of values for

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, one writes 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.

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

The group-like elements of a Hopf algebra do form a group.

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.

Every coalgebra is the sum of its finite-dimensional subcoalgebras, something that is not true for algebras.

Abstractly, coalgebras are generalizations, or duals, of finite-dimensional unital associative algebras.