Representation of a abstract-vector space?

Question

I have seen two definitions of a representation, one which says it is defined on a complex vector space (Fulton and Harris) and another which defines it on a real or complex vector space (source not publicly available). Why is there a restriction to these types of vector spaces and not more general vector spaces (i.e. including abstract vector spaces)?

Answer 1

If your question has to do with the representation of a group $G$, then in fact you can ask this question over any field $F$. If you are interested in complex representations of a group (as many physists are), then apparently you check those existing over $\mathbb{C}$; if you want to study over the reals $\mathbb{R}$, for other reasons, then you are interested in real vector spaces $V$.

Moreover there are cases where someone is interested in representations over fields of positive characteristic $p$, and study more abstract representations. In particular there is a whole branch in algebra called modular representation theory, which studies the representations of finite groups over this kind of fields, and where the cardinality of the group is divisible by $p$. In that case there is a quite fruitful theory and heavy algebraic machinery to sort out the situation.

EDIT

As the comment above says, the setup of representation theory is quite broad and you can ask for representations of groups over arbitrary rings with no particular structure too.

Answer 2

It depends on:

1) The object you want to represent

2) What known results about representation theory you want to use

About 1: It is very well possible to define a representation of a finite group on an abstract vector space over an abstract field $F$. The definition is just what you expect it to be: an homomorphism from the group $G$ to the group $GL(V)$ of $F$-linear transformations of $V$.

However, you cannot represent a complex Lie algebra (say) or more generally any kind of algebra (which by definition is itself a vector space over a field $F$) on a vector space over a completely different field $G$ (unless $G$ contains $F$ as a subfield). Again the definition of a representation of an algebra $A$ on a vector space $V$ is exactly what you would expect: an algebra morphism from your algebra $A$ to the algebra $End(V)$ of endomorphisms of your vector space, but this algebra morphism is in particular linear and hence requires that it is mapped between vector spaces (both $A$ and $End(V)$ are vector spaces with extra structure) over the same field.

Since Fulton and Harris want to look at representations of complex Lie algebras, it makes sense for them to only look at representations on complex vector spaces.

About 2: Things get considerably easier when you represent stuff on vector spaces over fields that are algebraically closed. For instance then you have a nice version of Schur's Lemma. The complex numbers are of course everybody's favorite algebraically closed field. In the other direction: representation theory of finite groups becomes considerably harder (or at least very different) when you use vector spaces over a field whose characteristic divides the order of the group. Representing finite groups on vector spaces over $\mathbb{F}_l$ for $l > |G|$ is not so different from representing them over $\mathbb{R}$ (everybody's favorite non-algebraically closed field) but when representing them over $\mathbb{F}_p$ with $p$ the order of some element in $G$ it becomes a whole different ball game, called modular representation theory. Of course this could both be a reason to avoid this situation or to embrace it, depending on your mathematical taste.