Why should we study derivations of algebras?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

Here’s a practical reason to look at derivations, from one algebraist’s perspective, broken down in a few steps.

1. Suppose you would rather look at the group of automorphisms of the algebra, instead of the derivations. There are good reasons to do that, even if what you are really interested is the algebra itself. Practically, you can sometimes translate a statement you want to prove about the algebra into a corresponding statement about the automorphism group and then apply different techniques to get your desired proof. Philosophically, this is an algebraic analogue of Klein’s Erlangen program, and that analogy is meaningful thanks to Galois descent for example (see the discussion in section III.1 of Serre’s book [2]).

2. But the group of automorphisms can be technically more challenging. For example, the octonions are an 8-dimensional algebra but their automorphism group is some 14-dimensional Lie group (manifold with a group structure) so maybe it’s hard to work directly with it.

3. So you may choose to simplify the problem by linearizing it, i.e., replacing the manifold with its tangent space. When you do that, you replace the automorphism group with its tangent space at the identity, which is the Lie algebra of derivations of the algebra. It is technically easier to work with, because for example it is easier to tell if two subspaces of a vector space are equal than if two submanifolds of a manifold are equal.

You don’t lose too much when you make this switch from looking at the automorphism group to the Lie algebra of derivations (the tangent space) because there are theorems saying that such-and-such a property holds for the identity component of the group if and only if some other property holds for the Lie algebra, see for example section II.7 in Borel’s book [1]. And historically this has been a very productive way of approaching the problem, since it led to things like the Killing–Cartan classification of simple Lie groups.

That’s the idea. Here is a technical detail that may or may not be relevant to your situation. The relationship between results for the derivations and results for the automorphism group is tightest over fields of characteristic zero. However, by the 1970s, say, it was clear that many of the theorems about automorphism groups or derivation algebras should be true over fields of all or almost all characteristics. But the technicalities involved in carefully proving theorems about the automorphism groups over fields of non-zero characteristic (along the lines of Grothendieck’s SGA3) are a substantial step up in difficulty and step down in power from the situation in characteristic zero. So it was very natural for researchers at that time to focus their attention on the derivations instead of on the automorphism group and simply exclude a handful of characteristics where the derivation algebras don’t tell the whole story.

References

[1] Borel, Armand, Linear algebraic groups, 2nd enlarged edition. Springer. (1991).

[2] Serre, Jean-Pierre, Galois cohomology. Transl. from the French by Patrick Ion, Berlin: Springer. x, 210 p. (1997). ZBL0902.12004.