Coalgebras

Last time we introduced the coproduct which is the essential ingredient of a co-algebra. How can we understand it? If we think of the product as a machine which eats two numbers and generates another one we understand the coproduct as the same machine working in reverse. A Xerox machine can be understood as a coproduct, but a coproduct can be understood not only a a cloning machine but as an action which breaks up an elements into sub-elements. For example a complex number can be decomposed into a real and an imaginary part and each of those are nothing but other kinds of complex numbers.

One funny example comes from shuffling cards: cutting a deck of cards in two is the coproduct, while putting it back together in all possible ways is the product. Renormalization techniques in quantum field theory generates coproducts. Here is a partial list of well studied mathematical examples. The coproduct is usually expressed with the symbol \( \Delta \) and the product is represented by the symbol \(m\).

The first (and most trivial) example come from group theory. Consider finite linear combinations of group elements:

\( kG = \{ \sum_{i=1}^n \alpha_i g_i | \alpha_i \in k, g_i \in G\}\)

\(\Delta g = g\otimes g\)

This is nothing but a basic cloning operation. A bit more complex example comes from polynomial rings:

\( \Delta (x^n) = \sum_{i=0}^n (n~choose~i) x^i \otimes x^j \)

\(m(x^i \otimes x^j) = x^{i+j}\)

Much fancier example come from the cohomology ring of a Lie group, or the universal enveloping algebra of a Lie algebra which gives rise to the so-called quantum groups which have major physical applications.

For now the question is: can we generate a coproduct given a product, and a product given a coproduct? The answer is rather surprising. The answer is yes in both cases for finite dimensional cases, but in general one can only generate a product given a coproduct.

Then can we have a mathematical structure which has both a product and a coproduct? If such a structure exists, it is called a bi-algebra and this respects a compatibility relation where tau is transposition of the terms in the tensor product. 

Let's take the group example. Start from the upper left corner with \( g_1 \otimes g_2\) and move it horizontally:

\(g_1 \otimes g_2 \rightarrow g_1 g_2 \rightarrow g_1 g_2 \otimes g_1 g_2\)

Then take it down, across and up and see you get the same thing meaning the diagram commutes:

\(g_1 \otimes g_2 \rightarrow g_1\otimes g_1 \otimes g_2 \otimes g_2 \rightarrow g_1\otimes g_2 \otimes g_1 \otimes g_2 \rightarrow g_1 g_2 \otimes g_1 g_2\)

Usually this kind of commutative diagram are fancy ways of expressing mathematical identities. For the polynomial ring the commutativity of the diagram means that this holds:

(m+n choose k) = sum over i, j with  i+j = k of (m choose i) (n choose j)

Also Hopf algebras are special kinds of bialgebras and no wonder they have major applications in combinatorics.

Next time we'll talk about Hopf algebras. Please stay tuned.