The n-Category Café

Thanks. My question about boundedness was a stupid one: of course, $H$ isn’t bounded! My excuse is that I was commenting fast, in a hurry to leave the office and get home. Serves me right :-)

So do you have an inkling on whether boundedness on $\Delta_2$ could replace continuity in Faddeev’s characterization (Theorem 2.4 in your lecture notes)?

I don’t have an inkling, I’m afraid. I’m overawed by people who speak a foreign language well enough to use words like “inkling”, but that doesn’t help.

If I seriously wanted to weaken the continuity condition in Faddeev’s theorem by a boundedness condition, I might warm up by learning more about Cauchy’s equation. I know that any measurable additive function $\mathbb{R} \to \mathbb{R}$ is linear, and that measurability can be weakened to “bounded on a set of positive measure”, but I’ve never read the proof of either of these theorems.

Incidentally:

I confess to never actually having properly read the book, despite having worked on the subject…

I definitely haven’t “read” Aczél and Daróczy’s book either. I don’t think it’s a book one reads so much as consults. At first I found it forbiddingly technical, and it’s true that the style is very particular, but it’s not so bad once you get used to it.

So, there are about three types of abelian group manifolds that locally look like $\mathbb{C}$: a maximal-rank discrete lattice subgroup can have rank 2 or 1 or 0; the rank 2 case is modeled by $\mathbb{C}$, (or because it doesn’t really care about complex multiplication, by $\mathbb{R}^2$) with the usual addition; and the rank-1 case is just as nicely $\mathbb{C}^{\times}$, the nonzero complex numbers with complex multiplication; and the rank 0 cases are the Elliptic Curves. As groups, they’re just tori, but they can have lots of different complex structures.

Anyways, the third family of formal group laws would be a power series $x +_E y = x + y + O(x y)$ expressing an elliptic group in some natural chart near the identity on an elliptic curve $E$. And as it happens, every formal group law over the rationals (or other field of characteristic zero) has its own logarithm over the same field. That might mean the distinction between them as formal group laws isn’t interesting, but the $ln_q$ in your notes’ examples does different analytic things than ordinary $ln$, so… I was just wondering.

It might even be a terribly silly wonder, to the extent that “logarithm of a formal group law” really can mean “any power series of the form $x + O(x^2)$”; but, say, taking a projective chart of a projective curve of genus 1 would narrow the playing field a bit.

Thanks!

There was some discussion in class about the word “deformed”, which I’d used in reference to these surprise entropies. People were asking to what extent this fit in with more mainstream uses of the word “deformation”. I confessed that it was just something I’d made up.

On the other hand, there is perhaps some justification. The multiplicative formula for the $q$-logarithm,

$$\ln_q(x y) = \ln_q(x) + \ln_q(y) + (1 - q) \ln_q(x) \ln_q(y),$$

gives rise to a multiplicative formula for the function $d_q \colon x \mapsto x \ln_q(x)$,

$$d_q(x y) = x d_q(y) + y d_q(x) + (1 - q) d_q(x) d_q (y),$$

which in turn gives rise to a multiplicative formula for the surprise entropy of order $q$,

$$S_q(\mathbf{p} \otimes \mathbf{r}) = S_q(\mathbf{p}) + S_q(\mathbf{r}) + (1 - q) S_q(\mathbf{p}) S_q(\mathbf{r}).$$

This is different from the multiplicative formula for entropy in my notes, but more resembles the kind of formulas in your comment.

The multiplicative formula for entropy in my notes was

$$S_q(\mathbf{p} \otimes \mathbf{r}) = S_q(\mathbf{p}) + \Bigl( \sum p_i^q \Bigr) S_q(\mathbf{r}).$$

This has the notable feature that the right-hand side is non-symmetric in $\mathbf{p}$ and $\mathbf{r}$ except in the “classical” case $q = 1$. So there are echoes of situations in deformation theory, where we have a classical algebra that’s commutative, but its deformations are all noncommutative.