group of characters

In our lecture notes, it says that the group of characters of a hopf algebra is the set of algebra maps from H to k.
The operation of this group is the convolution product which is related to the algebra’s coproduct. Do the characters not then also need to be coalgebra maps?
Since there is so close a relationship between the antipode and the group, it seems that the group elements would need to respect the co-product as well as the product…

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3 comments

  1. Just to make sure we talk about the same thing, what is the relationship you mention between the convolution product and the algebra’s coproduct?

    • briansdumb

      Convolution product is defined by taking the coproduct of some element, then applying each “convoluteer” to one side of the tensor and then taking the product.

  2. The short answer is the the word ‘character’ comes from representation theory, and characters of algebras are defined in general. So if we were to add some of the coalgebra structure to the definition of character, we would have to give it a new name (‘bicharacter or Hopf character’).

    The long answer is that it turns out to be interesting to study characters of Hopf algebras, using the classical definition of character. Similarly, it is useful to study the notion of module over a Hopf algebra. A module over a Hopf algebra is just a module over the underlying ring structure (as every algebra is a ring – forget the underlying field). It turns out that a character is equivalent to a module which is also one-dimensional over the underlying field.

    Of course, since an algebra has a base field, any module over an algebra is also a vector space over the ground field. So people have asked the following questions (assuming we work with a fixed algebra over a fixed field):

    1) When is the ground field a module over my algebra?
    2) Given two modules, when is their tensor product (over the ground field) another module?
    3) When is the vector space dual of a module also a module?

    It turns out that these questions are easy to answer if the algebra is a Hopf algebra – without changing the definition of module. Then the counit can be used to turn the ground field into a module, the coproduct can be used to define a module structure on tensor products, and the antipode can be used to give a module structure on duals of modules (see http://en.wikipedia.org/wiki/Representation_theory_of_Hopf_algebras for more info).

    Of course this doesn’t answer your question, but it does point out that sometimes it is fruitful to take a topic from algebra, and see what new information we can ‘discover’ when the algebra is a Hopf algebra, *without* changing the definitions of the classical object (in this case, modules or characters). Of course, there are notions like Hopf module, where we do require more than just an action.

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