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This is largely a question about terminology. I will introduce it with a simple example.

Consider a point $x$ on the real line and a function $f:\mathbb{R}\to\mathbb{R}$. Normally we say that $f$ is stationary at point $x'$ if $\left. \frac{df}{dx}\right|_{x'}=0$. We could also say that $f$ is stationary if $f$ is invariant to a small change in $x$. In slightly unusual notation, we could write this as $f(x'+dx)=f(x')$.

If (in this notation), $f(x'+dx)=f(x')$ then we also have $f(x'+Adx)=f(x')$ for any real number $A$. The transformations $x'\to a'+Adx$ thus form a group, which in this case happens to be equivalent to the reals under addition.

This obviously generalises to vector spaces, classical fields, etc. In each case we can say that something is stationary if it is invariant to some group of small/local/differential/infinitesimal transformations. It seems as if it might generalise further, to some more abstract definition of small/local.

My question is simply which (if any) of these is the correct term for a group of "small" transformations in the neighbourhood of some specific point. A secondary question is whether there is indeed some deep generalisation of what I've written above. The concept seems related to that of Lie algebras (which I haven't studied), but the motivation seems quite different and I have no idea if the concepts are actually the same.

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  • $\begingroup$ This is precisely the concept of an object being invariant under the action of a Lie algebra, and the motivation is precisely the same - Lie algebras abstract the notion of infinitesimal symmetries. $\endgroup$ – Qiaochu Yuan Jun 16 '14 at 4:09
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You would like to study invariance under (near-identity) infinitesimal transformations. One has the choice of studying the transforms, which have to be near the identity to produce invariance, or taking their logs and studying the resulting infinitesimal objects. This is the core algebraic content of Lie's theory of symmetry solutions of differential equations.

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