Hopf Algebras, Symmetric Functions and Representations

1.1. Motivation. Much of representation theory can be unified by considering the representation theory of associative algebras. Specifically, the representation theory of Lie algebras may be studied via the representations of universal enveloping algebras; the representation theory of finite groups studied via the representation theory of the group algebra; the representation theory of quivers studied through the representation theory of the path algebra; and so forth. The idea is that the representation theoretic aspects of a particular mathematical object is captured by some associative algebra that is, in a rather precise way, naturally related to the original object. The associative algebras arising in each of the above constructions, however, are much more structured and well-behaved than general associative algebras due to the representation theory underlying the original objects. To illustrate more precisely what we mean here, let A be any algebra over C—for convenience, we will work over the complex numbers, but for the most part, the base field will not be of concern—and consider two representations V and W of A. There are general constructions which we would like to perform on V and W and obtain new representations of A, namely, we would like to make sense of the sum representation V ⊕W, the tensor representation V ⊗W and the dual representation V∗. The sum representation V ⊕W is easy to make sense of even for general algebras A: for v⊕w ∈ V ⊕W, we can define an A-action by acting on each component separately, a · (v⊕w) = (a · v)⊕ (a ·w). It is not hard to check that this actually defines a representation on V ⊕W. The tensor and dual representations, however, are a bit more tricky. The issue with the tensor product, for instance, is that V ⊗W has an induced A⊗ A-action from the A-action on each of the components, but this does not automatically yield an A-action on V ⊗W. The naïve thing to do here is to attempt to define an A-action simply as above, by having A act on the tensor component-wise. This happens to work in the case where A = C[G] is a group algebra, but this does not work, say, when A is the universal enveloping algebra of a Lie algebra, as we have seen. One solution to this problem is to fix a map ∆ : A→ A⊗ A, called comultiplication, and have A act on V ⊗W through the map ∆. That is, we define an A-action on V ⊗W by