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Why is it called the Euler identity? Who calls it that? And are there noteworthy applications of it, by Euler or anyone else?
If you google it, then “Euler identity” is mostly used for other things: usually it’s $e^{\pi i} + 1 = 0$, but maybe for some people it’s $e^{i x} = \cos x + i\sin x$ (although most people call that Euler’s formula, I believe), or for other people maybe it’s $V - E + F = 2$. I think I object to the nLab calling this thing in the article “the Euler identity”, and I want to understand better what significance it’s being assigned by the authors of this article.
(It’s nice, but in my mind a bit on the trivial side, that spaces of homogeneous polynomials are eigenspaces of the derivation $x \frac{d}{d x}$. It leaves me wanting more.)
You’re right, it’s not known under the name “Euler identity” but under the name “Euler theorem for homogeneous functions”.
It’s important for me because I defined the notion of graded differential category in this paper and this identity plays a special role in this context (which is not discussed in this paper).
In the definition of a $\mathbb{N}$-graded codifferential category (the opposite of a $\mathbb{N}$-graded differential category), you have in particular that it is a symmetric monoidal category, a family of endofunctors $(!_{n}:\mathcal{C} \rightarrow \mathcal{C})_{n \ge 0}$, a unit natural transformations $I \rightarrow !_0 A$, multiplication natural transformations $(!_{n}A \otimes !_{p}A \rightarrow !_{n+p}A)_{n,p \ge 0}$ and a family of natural transformations $(d_n:!_{n+1}A \rightarrow !_{n}A \otimes A)_{n \ge 0}$.
The object $!_{n}A$ can be interpreted as a space of differentiable functions of some degree $n$ with coordinates in $A$ such that when you compose them, you multiply the degrees, the vectors are functions of degree 1 (these two items are realized by the structure of graded monad of $(!_{n})_{n \ge 0}$ which is also an ingredient in the definition of $\mathbb{N}$-graded codifferential category), when you multiply them, you multiply the degrees and the scalars are functions of degree $0$.
In the most natural examples, $\mathcal{C}$ is the category of vector spaces, $!_{n}A$ is really some set of smooth functions with coordinates in $A$ and the natural transformation $d_{n}$ acts like this: $f \mapsto \underset{0 \le i \le n}{\sum}\frac{\partial f}{\partial x_{i}} \otimes x_{i}$. If you follow this by the unit natural transformation on the right and then a multiplication, you obtain $f \mapsto \underset{0 \le i \le n}{\sum}\frac{\partial f}{\partial x_{i}}x_{I}: !_n A \rightarrow !_n A$ for every $n \ge 0$.
It turns out that in a $\mathbb{N}$-graded codifferential $\mathbb{Q}^+$-linear category, every $!_{n}A$ is the $n^{th}$ symmetric power of $A$ (a notion that you can define in any symmetric monoidal category) if and only if the above natural transformation is equal to $n.Id$ for every $n \ge 0$.
In a preprint “String diagrams for symmetric powers I” (not yet on ArXiv), I use these intuitions in a much lighter setting than graded codifferential categories to characterize the families $(S^n A)_{n \ge 1}$ of all symmetric powers of an object in a symmetric monoidal $\mathbb{Q}^+$-linear category: they are the families $(A_{n})_{n \ge 1}$ of objects which can be equipped with a family of morphisms $(\nabla_{n,p}:A_{n} \otimes A_{p} \rightarrow A_{n+p})_{n,p \ge 1}$ and a family of morphisms $(\Delta_{n,p}:A_{n+p} \rightarrow A_{n} \otimes A_{p})_{n,p \ge 1}$ such that $\nabla$ followed by $\Delta$ and $\Delta$ followed by $\nabla$ verify two simple identities which are reminiscent of the (higher-order) Leibniz rule and an higher-order “Leibniz identity”: $f^{(p)}.x^{p} = \binom{n}{p}.f$ for every $0 \le n \le p$ iff $f$ is an homogeneous polynomial of degree $n$ (in the case of univariate polynomials, which is really very simple). These two equations are expressed very nicely using string diagrams.
In a future preprint “String diagrams for symmetric powers II”, I will characterize the families of symmetric powers of an object in a symmetric monoidal $\mathbb{Q}^+$-linear category using only $(\nabla_{n,1}:A_n \otimes A_1 \rightarrow A_{n+1})_{n \ge 1}$ and $(\Delta_{n,1}:A_{n+1} \rightarrow A_{n} \otimes A_{1})_{n \ge 1}$, which turns out to be a bit more complicated and makes me introduce a rewriting system on these string diagrams in order to make the proof. I like it because it makes you understand facts about permutations, that are nicely represented using string diagrams.
It turns out that in a symmetric monoidal category enriched over commutative monoids which is not $\mathbb{Q}^+$-linear, there are several families of objects $(A_{n})_{n \ge 1}$ which verify the above conditions, but now they can be several things such as: the family of all symmetric powers of an object $A$, the family of all divided powers of $A$, the family of all exterior powers of $A$ if we are in $Vec_{\mathbb{Z}_2}$ and maybe other ones.
To sum up, I started being interested by this theorem of Euler for homogenous functions because it seemed useful in the context of differential categories and now I use this idea it to prove facts on symmetric powers and related functors in symmetric monoidal categories enriched over commutative monoids, where I use it as an additional condition on $\mathbb{N}$-graded bialgebras or similar algebraic structures. I don’t know if these considerations could be maybe useful in more complex matters about symmetric powers in positive characteristic such as for instance the characterization of all the natural transformations between symmetric powers functors in the paper “Generic Representations of the Finite General Linear Groups and the Steenrod Algebra III” by Nicholas Kuhn. In any case, I’m happy with what I’ve already done so far starting from this theorem by Euler and I’ve integrated it usefully in my intuition about graded bialgebras.
Thank you for your interest about this theorem of Euler and the reasons why it’s interesting to me.
Another note: this fact is mainly known in (chemical) thermodynamics and I actually learned it in a chemistry course.
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