I'm still very new to learning about Lie groups, something I find particularly confusing is the use of the word representation in the context of Lie groups. Sources I've checked online go quite far over my head and tend to be quite mathematical, so I thought I would try to ask about this term specifically.
When we say a "representation" of a Lie group, I interpret this as meaning that given the abstract set of elements and group structures, there are several "things" that the elements of the set could be while still satisfying the axioms of being that particular Lie group. For instance, when we say that $U(1)$ "is the circle group", and is the set of complex numbers of modulus $1$ with the group operation defined as complex multiplication, I find it strange that we are identifying something entirely abstract (a particular Lie group) with something specific (the complex numbers and complex multiplication). Am I correct in saying that there exist other sets of elements related by a different group operation that are another "representation" of the $U(1)$ Lie group? So that the elements of this other representation are in a sense isomorphic to the set of complex numbers with modulus $1$?
Another example would be the $SU(2)$ group, my lectures define this as the "special unitary group of $2\times2$ matrices with determinant equal to $1$". This again seems like we are defining this Lie group to be the set of matrices satisfying those axioms, but to me a group is something entirely abstract. So just like the $U(1)$ group I just talked about, do there exist other mathematical objects paired with another mathematical operation that is in the same sense isomorphic to the set of matrices defined above? And this alternative set of elements whatever they may be is just another "representation" of the $SU(2)$ Lie group?
I hope this question makes sense, I have tried to be as broad with it as possible, I'm not necessarily singling out the $U(1)$ or $SU(2)$ groups as confusing, but trying to understand the general idea behind this.