# Are group representations possible when the solution space is not a vector space?

As far as I understand, the motivation for using representation theory in high energy physics is as follows. Assume that a theory has some (internal or external) symmetry group which acts on a vector space. Then fields satisfying the theory will have to transform under some representation of that symmetry group, by construction.

What happens if we have some internal or external symmetry structure that is no longer acting on a vector space? The gauge group diffeomorphisms of general relativity spring to mind. Is there some more general 'representation' type theory which comes to our aid? And are there any examples of internal symmetries where this viewpoint is needed?

Apologies if this question is imprecise or flawed - I'm just starting to get my head around the foundations of the subject! Many thanks in advance!

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Don't the GR diffeos also act - $\frac{\partial {\bar{x}}^{\mu}}{\partial x^{\nu}}V^{\nu}(x)$ on a vector space - namely the space of sections of the tangent bundle? – twistor59 Feb 9 '13 at 14:27
assuming your group is a Lie group, there's always the adjoint representation on the corresponding Lie algebra, which is a vector space – Christoph Feb 9 '13 at 14:58

Let $G$ be a group, e.g. a finite group or a Lie group.

Then there exists the notion of a group action $G\times X\to X$, where $X$ is a set. The set $X$ does not necessarily have to be a vector space. It could e.g. be a manifold. And even if $X$ has vector-space structure, the group action could be non-linearly realized, i.e., a group element $g\in G$ is represented by a non-linear operator $T_g:X\to X$.

Non-linear realizations pop up all over the place in modern physics. For instance, in nonlinear realization of supersymmetry, or in nonlinear realization of the conformal group.

Example: Let the Lie group $G=GL(2,\mathbb{C})$ of invertible $2\times2$ matrices

$$\tag{1} A~=~\begin{pmatrix}a & b\\c & d \end{pmatrix}, \qquad \det(A)\neq 0,$$

act on the complex plane $\mathbb{C}$ (which, by the way, is a vector space) as

$$\tag{2} A.z ~:=~\frac{az+b}{cz+d}, \qquad (AB).z ~=~ A.(B.z)~.$$

In this way, matrices get non-linearly represented as meromorphic functions. The subgroup $SL(2,C)$ is the global conformal group in two space-time dimensions, which e.g. plays a fundamental role in the world-sheet description of string theory.

Finally, let us mention that in mathematics there exists a generalization of the notion of a $\mathbb{F}$-vector space, where the field $\mathbb{F}$ is replaced by a ring $R$. It is known as an $R$-module.

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I would like mention as an example, that for nonlinear differential equations, where the solution may not span linear vector spaces, we still capable of using groups of symmetry to solve them, as a reason to what you mentioned above. – TMS Feb 10 '13 at 9:00
@Qmechanic - thanks for your answer. I already know about general group actions, I was really just wondering whether there were any concrete examples of symmetry in physics where they were used. Could you possibly add some more to your answer, if there are any? Then I shall certainly accept! Many thanks! – Edward Hughes Feb 11 '13 at 16:47
I updated the answer. – Qmechanic Feb 11 '13 at 22:02
@Edward Hughes - here is an example of a non-linear realization of the symplectic group in the optics of Gaussian beams. – Stephen Blake Feb 11 '13 at 23:10