Cartan's Structure Equations VS Cartan's Method of Equivalence

M

Malkoun

There have been a number of posts on related questions, such as: Geometric interpretation of Cartan's structure equations, What is the geometric significance of Cartan's structure equations? and Maurer-Cartan structure equation derivation.

While my question is related, it is I hope a bit more specific. I am trying to learn some of Cartan's methods, and getting a little confused, because they all seem to be closely related, so that differentiating between them is difficult for me.

I would like to think of Cartan's structure equations this way. You start with an adapted co-frame and apply d, then express the results in terms of old data (the co-frame), but while respecting the Lie algebra. This gives 2 new things, the connection 1-form, and torsion. We then apply d again, and express the results in terms of "old" data (maybe while respecting an underlying Lie algebra).

I am trying to make sense of this. It seems that this is the same process than the one used in Cartan's equivalence method. Am I right? So torsion is the first "invariant", which could be used to compare 2 different geometric structures locally, while curvature would be the next "invariant".

But what is the relevant EDS perhaps? It seems that we are building something recursively, so I am a little confused. Perhaps we need to go to infinity, to see the whole structure, right? As in, using infinity-structures, like Urs Schreiber seems to be suggesting, in the second link above. Can someone please comment or answer my questions?

Edit: after some thinking, and reading a good chunk of Olver's book "Equivalence, Invariants and Symmetry", here is my current understanding of Cartan's structure equations. Let's say you have a Riemannian manifold $(M,g)$, and let $(\theta^i)$, for $1 \leq i \leq m$, with $m=\dim M$, be a smooth local orthonormal coframe. Applying $d$ to the coframe gives our first set of "invariants" (or perhaps I should write $O(m)$-invariants). That the first set of "invariants" is nothing but the Levi-Civita connection is the meaning of the first structure equations. We then apply $d$ a second time, and get a second set of "invariants". That this second set of invariants can be broken in 2 parts, the first quadratic in the Levi-Civita connection and the second one nothing but the curvature of $g$, is the content of Cartan's second structure equation.
 

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When an algebra isomorphism preserves positive involution

Let $A$ be a $K$-algebra where $K$ is a field with a unique ordering. We say a $K$-linear involution $*$ is positive if the map $A \to K$ via $a \mapsto tr(a^*a)$ is positive definite with respect to the ordering. Here, $tr(a)$ is the trace of the left multiplication map $a:x \mapsto ax$.

Suppose there is a $K$-algebra isomorphism $$\varphi: A \to B.$$

If there is a unique (up to isomorphism) positive involution in $A$ and $B$, can we say the involution must be preserved by the isomorphism $\varphi$: $$\varphi: (A, *_A) \to (B, *_B)?$$

Motivation:​


My question is motivated from representation theory. When $W$ is a real irreducible representation (irrep) of a finite group $G$, then $D:=End_G(W)$ is a division algebra by Schur's lemma. Additionally, Frobenius theorem on real associative division algebras implies that $D$ is isomorphic to either the reals $\mathbb{R}$, complexes $\mathbb{C}$, or quaternions $\mathbb{H}$. Let's call this isomorphism $\varphi$.

Suppose $W$ is endowed with an $G$-invariant inner product (i.e. $\langle g\cdot u, g\cdot v \rangle=\langle u , v \rangle$ for all $u,v \in W$ and $g \in G$). So, $W$ is an orthogonal irrep, and $D$ has the involution that acts as the adjoint corresponding to the $G$-invariant inner product. We know that $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$ all have involution. Furthermore, they are normed division algebras. The involution is the trivial map in $\mathbb{R}$, complex conjugate in $\mathbb{C}$, and the standard involution in $\mathbb{H}$ $\left((a+ib+jc+dk)^*=a-ib-jc-dk\right).$ We even do not need to define the involution in $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$ because the involution must be positive when we want to correspond involution to adjoint and there is a unique positive involution up to isomorphism in real central division algebras. My original question is how to prove that $\varphi$ preserves involution. Is this even true?

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copied image

now the problem is; in pose mode if i rotate or modify left shoulder bone, that right mirrored armor part is moving too as in image:

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I've tried to separate all object parts, then clear all parents from it, re-assign right shoulder bone as parent. but still it move with left shoulder bone as well as right shoulder bone.

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blender file:

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I have sown some garlic cloves in a pot (I only have a balcony, don't judge). And... something came out of the pot, but it's not garlic, and I'm very curious what it might be: Mystery plant Mystery plant, image two

I even mistook it for garlic initially, but it's completely inedible, and in fact, can barely be chewed through in any manner - the leaves are pretty tough. As far as I could tell by rummaging in the ground, there are no cloves to this beastie, and the roots seem to go quite far down. Google Lens identifies that as Chives, but I can guarantee you it's not (I actually do have chives, in a pot beside this one). In this case, you can see some actual garlic that caught on, chilling in the background, wondering what kind of neighbor is that. It definitely feels like a weed, but I still wanted to find out what it could be before it goes out.

Any educated guesses?

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Note: This question is for an ice cream machine WITH a compressor

Pretty much all recipes for ice cream/gelato call for completely chilling the mixture before starting to churn.

I get that for "passively" cooled machines (compressorless) it's important, so that you don't loose any of the pre-cooling you've done to the hardware.

Does it actually matter for a compressor-based machine? The machine can certainly cool it down without a hitch, but would it impact the final texture?

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My understanding of the Fiege-Fiat-Shamir protocol, sourced from "RSA and Public-Key Cryptography" by Richard A. Mollin is as below:

  1. A third party chooses two large primes $p$ and $q$, sharing $n = pq$.
  2. Alice has a secret value $s_a$ and shares a number $t_a = s_a^2 \mod n$.
  3. Alice picks a random $m \in \mathbb{N}$ and sends $w = m^2 \mod n$ to Bob.
  4. Bob picks a challenge $c \in \{0, 1\}$ and sends that to Alice.
  5. Alice computes $r = ms_a^c \mod m$ and sends that to Bob.
  6. Bob computes $r^2 = m^2s_a^{2c}$, and he can compare that to the publicly known $wt_a^2$, which should be identical if Alice does indeed know $s_a$. The security of this protocol is based on the fact that it is difficult to solve quadratic modular congruences with unfactored moduli.

My text elaborates that there is a method that an attacker can use to impersonate Alice with a 50% success rate, so this is done $a$ times and the probability that this works becomes $2^{-a}$.


If my understanding of this is correct, then one could use a similar protocol, but without the 50% chance of failure:

  1. As before, a third party selects $p$ and $q$, sharing $n = pq$.
  2. Alice picks a random $m$ and sends $w = m^2 \mod n$ to Bob.
  3. Alice sends $r = ms_a \mod n$ to Bob.
  4. Bob computes $r^2 \mod n$ and confirms that it's equal to $mt_a$.

Again, because neither $w$ nor $t_a$ are easy to compute, this seems secure.

I assume that this doesn't work. Why not? Why does Fiege-Fiat-Shamir require that Bob send a challenge?

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What common proverb does this string refer to?

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What common proverb is referenced by this three-character string?

ᚷᛖᚨ
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