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There is this abstract notion of a derivation, which really only cares about the property

$$D(ab)=aD(b)+D(a)b,$$

where $a,b$ are elements of some algebra. This only tangents the ideas, which lead to $\frac{\text d}{\text d x}$ for functions on the real line.

I wonder if there is such a thing as as differential equations in abstract algebra? I guess I can just write down an equation $D(aD(ab))=abaa$ or $D(a)=-ca$ for some algebra and some $D$ and try to figure out if there are actually elements, which satisfy this relation, but I wonder if peolpe are actually doing such things and what their insights turn out to be. Are there investigations of e.g. initially physically motivated equations in terms of this abstract concepts?

The only related variant I can think of are equations of Lie-Groups, however, it seems these can be expressed in terms of the usual derivative as well (since they also carry the manifold structure), so it's not really something new.

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3 Answers 3

Your questoin is not clear. More or less any differential equation is of the form you mentioned, but you probably don't mean those. But in any case you need to specifiy which derivation $D$ you mean for your equation to make sense. Even if you try to make exotic examples of derivations, the equation that you write down is likely to give you a differential equation anyway, though maybe not the same as if you take $D$ to stand for ordinary differentation. If you are asking if there are algebraic structures where one writes down equations in which there occur operations that are derivations, then the answer is yes. For instance in a Poisson algebra any Poisson bracket operation $\{x,\cdot\}$ is a derivation.

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Okay, ya well a Poisson algebra can also be represented in terms of ordinary partial operators and they act on functions over real. At least in the example I have seen them. Many ideas really are only abstractions from what people know from analysis. I probably should focus on the question of algebras together with derivation operatons, which really are not isomorphic to some function space in differential geometry. –  NikolajK Mar 27 '12 at 14:11

I once investigated this equation on the reals. In other words, D becomes a function. It is unique up to the base:

$D(x) = x.log(|x|)$ in an arbitrary base.

It then extends to an operator on functions as well:

$D(f) = f.log(|f|)$ (in fact for each x, a different base can be taken)

The inverse operator (algebraic integration) is not unique, because the function $D:x \to x.log(|x|)$ is not monotonous. Maybe there are solutions?

Hope this helps a bit.

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I know two kinds of results in differential algebra:

  1. (My favorite) Take any function to show that can be integrated in finite terms: for example $\int e^{-x^2}$ is not integrable in finite terms, that is, the integral cannot be expressed with rational, exponential, logarithmic or trigonometric functions. There are algorithms to integrate in finite terms or to decide it cannot (Brookstein, for example). In this way Differential Algebra is like resolution of algebraic equations with radicals.
  2. Similarly, take a ODE and show that can be solved in terms of elementary functions or extensions by integrals of them. Here my reference is very old: Ritt (1950). In this way Differential Algebra is a type of Galois theory.

Obviously Differential Algebra is more.

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