TITLE: Why should we study derivations of algebras?
QUESTION [14 upvotes]: Some authors have written that derivation of an algebra is an important tools for studying its structure.  Could you give me a specific example of how a derivation gives insight into an algebra's structure?  More generally:  Why should we study derivations of algebras?  What is the advantage of knowing that an algebra has a non-trivial derivation?

REPLY [7 votes]: Here are some algebraic aspects why one could be interested in derivations.
First, derivations actually appear pretty prominently in algebraic geometry. For a ring homomorphism $R\to S$, one can consider $R$-linear derivations on $S$, with image in some $S$-module $M$. There is a universal such module, the module of Kähler differentials $\Omega_{S/R}$, with a universal derivation $d:S\to\Omega_{S/R}$. This is a fairly important object to study in algebraic geometry, e.g., in the definition and study of smooth varieties, or in the algebraic versions of de Rham cohomology. (A chapter on derivations can be found in any algebraic geometry textbook.)
On the other hand, and a bit more specialized, there are also reasons to study locally nilpotent derivations on algebras. Since locally nilpotent derivations are related to actions of the additive group $\mathbb{G}_a$ on varieties, the locally nilpotent derivations also play a significant role in the study of quotients of $\mathbb{G}_a$-actions. In particular, (as found on the Wikipedia page) some counterexamples or Hilbert's 14th problem arise as kernels of locally nilpotent derivations. (survey paper here)
Locally nilpotent derivations are also related to various open questions on the structure of affine space $\mathbb{A}^n$ (like the Zariski cancellation problem, see e.g. the Makar-Limanov invariant which can distinguish affine space from some affine varieties diffeomorphic to it). There is a whole book about the algebraic theory of locally nilpotent derivations:

G. Freudenburg. Algebraic theory of locally nilpotent derivations. Encyclopaedia of Mathematical Sciences, 136. Invariant Theory and Algebraic Transformation Groups, VII. Springer-Verlag, Berlin, 2006.

REPLY [2 votes]: The two examples which I can give are the following. One is the algebra of octonions. The derivations of this algebra form a 14-dimensional Lie algebra. From this fact you can conclude that the automorphism group of the octonions is 14-dimensional. An automorphism is obtained from a given derivation by taking the $exp$ function. The exponential function doesn't work over finite fields. There are formulas for derivations of octonions e.g. 
$[L_x,L_y]+ [L_x,R_y]+[R_x,R_y]$ is a derivation for every pair of octonions $x,y$. In this formula $L_x$ and $R_x$ denote left and right multiplication by the octonion $x$.
You can find more in Schafer's book or Barton-Sudbery 
https://arxiv.org/abs/math/0203010.
We can define the Jordan algebra $h_3(\mathbb O)$ of symmetric matrices of octonions. In this algebra $[L_x,L_y]$ is a derivation. The algebra of derivations is 52-dimensional in this case. It is the $f_4$ Lie algebra.
I wonder whether the $e_6$, $e_7$ and $e_8$ Lie algebras can be obtained as derivations of some algebra.