The Hopf Algebra

Continuing the discussion, a bialgebra is a structure which is both an algebra and a coalgebra subject to a compatibility condition. A Hopf algebra H is a bialgebra with yet another property: the antipode. The antipode is a map from H to H and is usually named S.

If the bialgebra is a graded space, then the antipode comes for free and by an abuse of notation people call bialgebras Hopf algebras.

The antipode must be compatible with the existing structures of multiplication and comultiplication and so the diagram bellow commutes.

One thing I forgot to mention last time is that the unit "u" has a dual: the counit \( \epsilon\) which maps elements from the algebra to the field \( k \). Most of the time the counit maps everything to zero and occasionally to one.

Let us verify the commutativity of the diagram with our friend \(kG \). Here the antipode is the group inverse: \( S(g) = g^{-1}\):

Start from the left side: \( g\) 

Moving up and right: \( g \rightarrow g\otimes g\) then moving to the right and down: \( g \rightarrow g\otimes g \rightarrow\ g^{-1} \otimes g \rightarrow 1_H\) 

Now move from left to right on the middle line. If \( \epsilon \) maps all elements to \(1_K \) we have:

\( g \rightarrow 1_K \rightarrow 1_H \) and the diagram commutes. 

For a graded bialgebra the antipode is given by the following explicit formula:

\( S = \sum_{n \geq 0} {(-1)}^n m^{n-1} {\pi}^{\otimes n} {\Delta}^{n-1}\)

where \( \pi = I - u \epsilon \)

Next time we will see Hopf algebra application to renormalization in quantum field theory. If you want to read about Hopf algebras, the standard book is a 1969 book "Hopf Algebras" by Moss Sweedler who is known for the so-called Sweedler notation. Personally I do not like the style of the book because you get lost into irrelevant details and miss the forest because of the trees, but it is a good reference.