[MSE] Help on a special kind of functional equation.

[MphLee]

Hi, I know this is not directly related to tetration. It is related to iteration theory and solving functional equations.

https://math.stackexchange.com/questions...alpha-beta

The question is the following

Mind that here x,y can be thought to be numbers. But the crucial part is that , and are intended to be matrices, formal powerseries or functions (non commutative objects in general) under composition.
I ask also here because with JmsNxn deep familiarity with iterated composition (in the question i rendered the Omega notation as a product to not scare away normies) I could get some knowledge.

[JmsNxn]

Is this some kind of semi-direct product? Jesus; that would be cool...

Let me think on this... I'll see if I can think of anything.

Is really supposed to be an element of ; rather than necessarily a conjugation?

........

Update:

So I'm not certain I can answer this; but, did you understand when I introduced the congruent integral (if you read the paper)? This is a different form of the compositional integral that is in a "modded out space". I can reduce (I think); your question into the abelian case; but I need to know how much of the congruent integral I should explain.

Also; what the solutions are referred to as is homormorphisms of the semi-direct products between two groups.

In this case you are taking an inner automorphism . Then you are constructing what is typically written,



And you are looking for



There's a word for this; I can't remember it exactly.

Also, this forum doesn't have the best latex implementation; so should actually be the symbol ; which is more angular.  It should look more like this 
.....

I'll explain this better tomorrow. Long night; but it's at least SOMETHING like this. Nonetheless; your answer lies in semi-direct products.

[JmsNxn]

I finally can answer your question. I was a little confused about the details before; but it's definitely a semi-direct product.


Let's first of all, ignore the function . Let's write a product,



And call this group,



Where . Where is the group of isomorphisms of .

Now; there are many choices of (something to do with Euler's phi function will be involved in the estimate of how many). Now when you write , you are choosing an isomorphism of ; let's call this . We can do this because we're only going to care about in the final result. And this is just considering at an implicit level in the preimage and considering it equivalent.

Now, here is where I wasn't making any sense before. You wrote your equation backwards from the usual semi-direct product. The right way I should've written; which I apologize for saying. Is that it's,



You are now looking for projections,



I fucking knew it was semi-direct products! Took me a while to think about it though...

Regards, James. I hope this helps.

---

Long story short; you have a lot of group theory at your disposal, Mphlee. May I recommend dummit & foote.

[MphLee]

Yes!
That's very promising. I'll have to work hard on this. Thank you so much! I believe that it is the solution: I just need to rephrase it, play with it a bit and prove it with my hands.

This problem popped out at the intersection between the abstract Jabotinky foundation and superfunction-spaces. Can't wait to find time to work on the details.

Thank you again.... nobody there on MSE even had the kindness to drop a single keyword. For group theorist this should be the abc... for sure my question is not well written, my English su*x, but I don't feel is a bad question.

[JmsNxn]

I think your question was written very well. It was probably more suited to Mathoverflow though; stack exchange tends to be mostly for undergraduate homework problems. But even then, the community can be a tad abrasive; and they always poke holes in technicalities while ignoring the larger picture. I deleted my stack exchange accounts a long time ago; bored by how little they actually help. Sometimes, you have to be left to your own wits .

I suspect though, no one saw that it was a semi-direct product... just a weird one. Only reason I saw it was because I was thinking about bullet notation. And I was initially trying to give an example of your function.

Regards, James