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?
[Math] we study derivations of algebras
derivationsra.rings-and-algebras
Best Answer
I think you want an answer from a pure algebraist, and I would be very interested in seeing such an answer as well. However, as a functional analyst, there are some things I can say to the question of why derivations are important which I find very convincing.
The basic point is that differentiation is a derivation. Most simply, the map $D: f \mapsto f'$ from the smooth real-valued functions on $\mathbb{R}$ to itself. This map satisfies the derivation identity $D(fg) = fD(g) + D(f)g$. More generally, differentiation along a vector field is a derivation from $C^\infty(M)$ to itself, for any smooth manifold $M$. The exterior derivative is another example; this time the target space is not $C^\infty(M)$ itself but a bimodule over it.
If you're willing to consider derivatives which are not continuous but merely measurable, then a natural domain of the differentiation operator is the set of real-valued Lipschitz functions. These all have derivatives in $L^\infty$. Of course we need to assume a metric now, in order for the notion of being Lipschitz to make sense.
In fact there is a strong connection between metrics and derivations. Theorem: let $X$ be a $\sigma$-finite measure space and $\delta$ a derivation from a dense subalgebra of $L^\infty(X)$ to a bimodule over $L^\infty(X)$. Given a natural continuity condition and the right notion of "bimodule", we can conclude that there is a metric on $X$ such that the domain of $\delta$ is precisely the set of bounded Lipschitz functions on $X$, and the norm of $\delta(f)$ is the Lipschitz number of $f$.
Here's another connection between metrics and derivations. Let $X$ be any metric space equipped with a regular Borel measure. Define $\Omega(X)$ to be the set of derivations from ${\rm Lip}(X)$ to $L^\infty(X)$ which satisfy a natural continuity condition. If $X$ is a Riemannian manifold then $\Omega(X)$ will be the set of bounded measurable 1-forms. But this construction makes sense for arbitrary metric spaces! It trivializes in many cases, but there is a large array of non-manifold metric spaces for which this construction is nontrivial and interesting: sub-Riemannian manifolds, rectifiable sets, Hilbert cubes, various kinds of fractals, etc. We have a first order exterior derivative in all these cases.
I could go on, but I'll stop here. These topics are covered extensively in the second edition of my book Lipschitz Algebras, which is now in press.