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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Ro 14 September 2024


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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?

problem with armature after mirroring part of object

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