Editing Coalgebra (section)

== Informal discussion ==
One frequently recurring example of coalgebras occurs in [[representation theory]], and in particular, in the representation theory of the [[rotation group]]. A primary task, of practical use in physics, is to obtain combinations of systems with different states of [[angular momentum]] and [[Spin (physics)|spin]]. For this purpose, one uses the [[Clebsch–Gordan coefficients]]. Given two systems <math>A,B</math> with angular momenta <math>j_A</math> and <math>j_B</math>, a particularly important task is to find the total angular momentum <math>j_A + j_B</math> given the combined state <math>|A\rangle\otimes |B\rangle</math>. This is provided by the [[Clebsch–Gordan coefficients#Tensor product space|total angular momentum operator]], which extracts the needed quantity from each side of the tensor product. It can be written as an "external" tensor product

:<math>\mathbf{J} \equiv \mathbf{j} \otimes 1 + 1 \otimes \mathbf{j}</math>

The word "external" appears here, in contrast to the "internal" tensor product of a [[tensor algebra]]. A tensor algebra comes with a tensor product (the internal one); it can also be equipped with a second tensor product, the "external" one, or the [[Tensor algebra#Coproduct|coproduct]], having the form above.  That they are two different products is emphasized by recalling that the internal tensor product of a vector and a scalar is just simple scalar multiplication. The external product keeps them separated. In this setting, the coproduct is the map
:<math>\Delta: J\to J\otimes J</math>
that takes
:<math>\Delta: \mathbf{j} \mapsto \mathbf{j} \otimes 1 + 1 \otimes \mathbf{j}</math>
For this example, <math>J</math> can be taken to be one of the spin representations of the rotation group, with the [[fundamental representation]] being the common-sense choice. This coproduct can be [[lift (mathematics)|lifted]] to all of the tensor algebra, by a simple lemma that applies to [[free object]]s: the tensor algebra is a [[free algebra]], therefore, any homomorphism defined on a subset can be extended to the entire algebra. 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 <math>\mathbf{j}</math> must be kept in sequential order during products of multiple angular momenta (rotations are not commutative).

The peculiar form of having the <math>\mathbf{j}</math> appear only once in the coproduct, rather than (for example) defining <math>\mathbf{j} \mapsto \mathbf{j} \otimes \mathbf{j}</math> is in order to maintain linearity: for this example, (and for representation theory in general), the coproduct ''must'' be linear. As a general rule, the coproduct in representation theory is reducible; the factors are given by the [[Littlewood–Richardson rule]]. (The Littlewood–Richardson rule conveys the same idea as the Clebsch–Gordan coefficients, but in a more general setting).

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