Do Algebra Representations Impact Theoretical Physics and Pure Mathematics?

[tgt]

1. Can an algebra have an infinite number of non isomorphic representations?

2. Can an algebra have two different representations where one is irreducible and the other is reducible?

3. In general, is it easy to come up with a representation of an algebra? If so then is there a preference for one representation then another?

 

[matt grime]

1. Yes: let k be the trivial representation, then k, k+k, k+k+k, k+k+k+k, etc are non-isomorphic

2. Yes, again, trivially: see 1.

3. I have no idea why one representation would be more preferable to another, but in general, beyond the obvious reps, it may be hard to construct even the simple modules explicitly. The indecomposable ones might not even be able to be written down meaningfully - this is called wild representation type.

 

[ThirstyDog]

3. Generally I have found the main preference is given towards irreducible over those which which are not (this includes indecomposable).

To add a little more for the construction of reps, there are those where the algebra is considered its own module. Also if you have a coproduct then it is possible to build different reps from a single rep (this is different from just taking the direct sum).

 

[matt grime]

Who has said what 'preferable' means? No one. It is usually more important to describe the irreducibles since all modules are extensions of them. I don't see how it is remotely consistent with the English language to say that makes one prefer them.

Direct product and co-product agree (in module categories - i.e. abelian categories) for finitely indexed sums and products. It's only when you have infinite indices that they are different. One also has tensor products of representations (possibly one should restrict to algebras with a co-multiplication).

 

[ThirstyDog]

The reason which I consider irreducible preferable is that in papers which I have read it is generally easier to classify the irreducible reps compared to the indecomposable. Merely that the preference is given to those which are easier to deal with. This is me coming from a more constructive side of representation theory rather than dealing with the more abstract approaches - thus preferable is referring to the focus of those in my field.

Also for coproduct I am referring to what you call comultiplication (I have spent more times with Hopf algebras then categories)

 

[matt grime]

Surely if it is easier to find the simples, then it is preferable to find all the indecomposables (if that is at all possible)? Sorry, it's just that I find "preferable" to be a very odd choice of word. In the situation you describe, 'necessary' seems better.

On that particular topic, it is trivial to find all the simple modules for a p-group in characteristic p, for example (there is exactly 1 simple in each case: the trivial rep). And very interesting (if very hard) to work out some theory for the other modules: C_p is easy, C_2 x C_2 is quite easy, C_p x C_p is a little harder, and after that it's complicated.Some people would say that finite dimensional (or finitely generated) modules are preferable as they're more 'natural' I suppose.

 

[ThirstyDog]

I guess it depends on what your area is... I am not that deeply involved in the theory of modules and representations thus find it useful to deal with 'nice' representation to get something concrete. For example generating representations of the braid group.

Can you give tell me of any simple papers or books which deals with representations with a category based approach? Too many times I have been to rep theory talks and been confused by those AR-quivers.

P.S. I do have a tendency to use more creative language than I need to when not dealing with strict mathematical question.

 

[matt grime]

The book for group algebras is Benson's Cohomology of finite groups vol I. A more useful reference for the same topic is Alperin's Local representation theory.

 

[tgt]

Well, given that there are many different representations for a single algebra. Would the mathematician usually want to deal specifically with one or many different ones? If one then what is the reason for choosing that one? In that situation, clearly the mathematician prefers that one compared to the other ones.

i.e there are infinitely many different fractional forms of 0.5 such as 1/2, 2/4, 3/6, 4/8 ...
But on most times I'd prefer 1/2.

So in that sense, it seems like the irreducible reps are the preffered one.

 

[tgt]

1. Yes: let k be the trivial representation, then k, k+k, k+k+k, k+k+k+k, etc are non-isomorphic

So k is the vector space spanned by one element?

 

[matt grime]

You seem to have extrapolated a personal and subjective opinion about fractions into one that representation theorists
almost surely don't share in any way.

Irreducible representations are the most important in a meaningful sense: every module is built from them.

 

[tgt]

Just on the topic of rep theory. It seems that it doesn't intrduce many new things. i.e unlike groups there are no new axioms associated with rep theory. And also if we look at just group reps, its just a homo. from a group to the general linear group which doesn't seem like a 'big' idea. But it seems to be a huge area. Why?

 

[matt grime]

If you want a major application of it (to pure maths) then the classification of finite simple groups springs to mind. Without representation theory you wouldn't have that.

There are some who would argue (and I would agree) that groups are only interesting (or important) when they act on something. Remember the axioms didn't come first - concrete groups did. Representations are the actions of groups on vector spaces, which are ubiquitous in mathematics and physics. Without representation theory you might not have, say, the Standard Model in physics.