The word "representation" in the context of Lie groups

Question

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.

I think this question may be better asked on Maths.SE. However, it appears you simple want to know the definition of a representation, in which case Wikipedia is pretty good. — jacob1729, Apr 28 '20 at 14:15
Even the Wikipedia section you've linked is quite mathematical, I'd prefer to work on the mathematical definition after I'm clear on the general concept. This might be better on MathSE though you're right. — Charlie, Apr 28 '20 at 14:41
That's fair, if my question doesn't make a lot of sense I might just be stuck at the starting gate on this without really knowing where to go. Your answers to related questions on representations are actually very helpful, I will continue to read those for now. — Charlie, Apr 28 '20 at 15:49
In principle given a group $G$ and a faithful representation in a Vector space $V$, i.e. an injective map $\rho:G\rightarrow GL(V)$ satisfying $\rho(g*h)=\rho(g)\cdot\rho(h)$, then the image $\rho(G)\subseteq GL(V)$ of the whole group $G$ can be identified with $G$ itself. Is that what your question is about? There are ways to define a Lie group in an abstract way. For example a simply connected Lie group can be defined by its abstract Lie algebra. — Johnny Longsom, Apr 28 '20 at 16:06
It's a bit hard to pinpoint where my confusion is, apparently my description of a representation outlined in the question is wrong. @JohnnyLongsom how could I understand the set of 2D rotation matrices as a representation of the Lie group $U(1)$ in terms of what you've written. Would $V=\Bbb R^{2}$ and $G=U(1)$? — Charlie, Apr 28 '20 at 16:47
@ChiralAnomaly I'm not really sure at this point, my description of "representation" in the question appears to be way off so I've actually ended up confusing myself more, I probably just have to keep reading for a bit until I can construct a more specific question. Thanks for your help though. — Charlie, Apr 28 '20 at 16:48
@Charlie On an abstract level the groups $U(1)$ and $SO(2)$ are the same in the sense that there is a bijective map $\psi:U(1)\rightarrow SO(2)$ that respects the group operation: $$\psi: e^{i\varphi}\in U(1)\mapsto \left(\begin{matrix} \cos\varphi & \sin\varphi \ -\sin\varphi & \cos\varphi \end{matrix}\right).$$ By enlarging the image set to $GL(\mathbb{R}^2)$ this map defines a 2-dimensional real ($V=\mathbb{R}^2$) representation of $G=U(1)\cong SO(2)$, but this enlargment is only to fit the definition with $GL(V)$. — Johnny Longsom, Apr 28 '20 at 19:39
@JohnnyLongsom I'm kind of surprised to see that $U(1)$ and $SO(2)$ are isomorphic, does that mean at the abstract group level they are both just a continuous set of elements with a group operation and topology/atlas to give the manifold/group structure, and it's only when we choose a particular representation that they become distinct? At the abstract group level how is the "orthogonal" condition imposed on the $SO(2)$ group? — Charlie, Apr 29 '20 at 11:52
@Charlie At the abstract group level there's absolutely no difference between $U(1)$ and $SO(2)$. One of them has the word "unitary" in its name and one has the word "orthogonal", but these are just crutches to help us remember what the groups are. — knzhou, May 04 '20 at 21:10
@knzhou And that's because of the isomorphism between them right? If that's the case, at the same level what is the difference between say $U(1)$ and $SU(2)$? Or some other matrix group? — Charlie, May 04 '20 at 21:34
@Charlie Well, they're different as groups. For example, $U(1)$ is one-dimensional while $SU(2)$ is three-dimensional. $U(1)$ is abelian while $SU(2)$ isn't. And so on. — knzhou, May 04 '20 at 21:40
@Charlie If you're getting very confused with Lie groups, I recommend backing up and relearning everything in the context of finite groups. For example, consider the finite group ${-1, 1}$ under multiplication. This is in fact the same group as ${\text{even}, \text{odd}}$ under addition. And both of these are the same as the abstract group with two elements $e$ and $a$ where $e$ is defined to be the identity element, and $a * a = e$ where $*$ is the group operation. — knzhou, May 04 '20 at 21:42

score 4 · Accepted Answer · edited Apr 04 '22 at 20:23

4

To a physicist, a "group representation" is a set of matrices (i.e elements of ${\rm GL}[{\mathbb F}]$ where $\mathbb F$ is a field, often ${\mathbb F}={\mathbb C}$) having the same multiplication table as the abstract group. The representation is faithful if different group elements correspond to different matrices. To a mathematician the representation is the map $D:G\to {\rm GL}[{\mathbb F}]$ with $g\mapsto D(g)\in {\rm GL}[{\mathbb F}]$ rather than the set $\{D(g)\}$ of matrices. This difference in usage is occasionally confusing. This confusion is also enhanced because physicists often refer to the vector space on which the matrices act as the "representation."

edited Apr 04 '22 at 20:23

Urb

2,608

answered Apr 28 '20 at 16:09

mike stone

52,996

Ok this helps, when we say the set of 2D rotation matrices are a representation of the Lie group $U(1)$, what corresponds to "the abstract group", is this the Lie group $U(1)$? If this is the case, why for instance on Wikipedia are we defining the group $U(n)$ as the set of $n\times n$ unitary matrices if these matrices are just one representation? – Charlie Apr 28 '20 at 17:01
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The $n\times n$ unitary matrices are *defining representation" of ${\rm U}(n)$. That that case the map in the mathematicians' definition is just the identity map. Similarly the abstract ${\rm U}(1)$ has a one -dimensional defining represenatation $e^{i\theta}$ and non-faithful represenations $e^{i\theta}\mapsto e^{in\theta}$, $n\in {\mathbb Z}$ as well as higher dimensional reducible such as the 2D rotation matrices representations. – mike stone Apr 28 '20 at 17:19
I will have to read up more on reducible vs irreducible representations, but what you've said makes a lot of sense so thank you. Just in the case of $U(1)$ you're saying that two possible representations of the Lie group $U(1)$ are the set of 2D rotation matrices and what is effectively the set of 1D complex matrices? And that the vector spaces on which they act are $\Bbb R^2$ and $\Bbb C$ respectively, which are sometimes referred to as the "representation" themselves? If so, how do we impose the restriction that the modulus of the entries of the complex case be $=1$? – Charlie Apr 28 '20 at 18:23

The word "representation" in the context of Lie groups

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