It is common that You have some interesting object (set, group, algebra or something, whatever) which has certain properties, structure etc. You may try describe it in pure algebraic way. Sometimes You cannot find representation different from this You are working on.
How to distinguish its algebraic properties form properties arising from particular representation? Is there any systematic procedure?
Is there only trivial answers possible: "find another model/representation", or "every object may be regarded as example of different algebraic structures"?
Or maybe it is enough to describe something without notions from representation ( finite dimensionality, matrix indexes etc)?
Example: suppose You have algebraic structure given by set M with some operations on it. It have representation in matrix algebra. Suppose that this representation has following properties:
- for every matrix M from representation $det(M) = 1$
- for every matrix from representation $Tr(M) = 0$
- whole representation is a vector space of dimensionality n with basis given by set of n independent matrices
- there is matrix S in representation for which $S^2=id$
- for certain matrices A and B C, there is relation AB^2 - C = C^2
If we regard its as a group only property 4 will be representation independent. But if we talking about vector space obviously property 3 is crucial.
We have some freedom to choose what is important: so if we are talking about abstract group property 3 may be called particular property of representation, whilst when we are in vector spaces, property 4 may be particular. Property 2 may be called fundamental if we are talking about "matrix groups" etc. What about property 5? Is it important?
Suppose we are able to wrote relations in a way which is pure algebraic ( for example for matrices we may wrote relations which not must be narrowed to matrix operations, we may interpret it as general algebraic property of more general object, as relation $S^2 =id$). And it lead to some interesting conclusions. How to be sure,property we are research is not only property of chosen representation? What with property 5?
Of course we may regard the same object as a group and as a vector space simultaneously. In this way however certain properties may be considered as detached one from each other. Is this the only way?