Derivative of the Ad map on a Lie group

In summary, the formula for Ad-map holds for any smooth curve on a Lie group. It is used as a definition of the ##ad##-map in the book "Dynamical systems and geometric mechanics".
  • #1
eipiplusone
10
0
Hi,

let ##G## be a Lie group, ##\varrho## its Lie algebra, and consider the adjoint operatores, ##Ad : G \times \varrho \to \varrho##, ##ad: \varrho \times \varrho \to \varrho##.

In a paper (https://aip.scitation.org/doi/full/10.1063/1.4893357) the following formula is used. Let ##g(t)## be a smooth curve on ##G##, with ##\frac{d}{dt}|_{t=0} g(t) = v##. And let ##u## be some arbitrary element in ##\varrho##. Then,

$$\frac{d}{dt}|_{t=0} Ad_{g(t)} u = ad_{\frac{d}{dt}|_{t=0} g(t)} u = ad_{v} u .$$

I know that this identity holds for Matrix groups, but the present setup is a general Lie group.

Furthermore, in the book "Dynamical systems and geometric mechanics", the above property is actually used as a definition of the ##ad##-map, for any ##v \in \varrho## and any curve ##g## with ##g'(0) = v##.

Any hints as to why the formula is true would be greatly appreciated.
 
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  • #3
(hopefully, the equations display now).

I don't see anything about the Ad-map in Pantheon-link - am I missing something?

Regarding Ado's theorem. As I understand it, it can be used to show that any finite dimensional Lie group is locally isomorphic to a Linear group (a matrix group). I guess that might allow us to "transfer" the formula from the Linear group (on which it holds) to the abitrary Lie group. I will think about that. If you have any hints as to how the argument would go, I would love to hear them.
 
  • #4
eipiplusone said:
I don't see anything about the Ad-map in Pantheon-link - am I missing something?
##\mathcal{L}_X(Y) = \operatorname{ad}X (Y) = [X,Y]##.

The most important formula is ##\operatorname{Ad} exp (\rho) = exp (\mathfrak{ad}\rho)##.

##\operatorname{Ad}## is induced by the conjugation in the group, leading to a conjugation with group elements on its tangent space, which if differentiated results in the left multiplication ##\mathfrak{ad}## in the Lie algebra.

You don't need Ado, I just mentioned it to emphasize that "not a general linear group" is relative. Of course there are local Lie groups as well, which do not naturally allow a matrix representation. Here's a nice example:
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/which can also serve as an easy to calculate example of the curves ##g(t)## you mentioned.

If you really want to dive in the subject, then I recommend Varadarajan's book on Lie groups. But for a quick look, the explanation of Lie derivatives and the examples there will do.
 
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1. What is the Ad map on a Lie group?

The Ad map on a Lie group is a function that maps elements of the Lie group to elements of its corresponding Lie algebra. It is defined as the derivative of the conjugation map, which is a group automorphism that sends an element to its conjugate by another element.

2. What is the purpose of the Ad map on a Lie group?

The Ad map on a Lie group is used to study the local structure of the Lie group. It allows us to understand how the group behaves near the identity element, which in turn provides insights into the group's global behavior.

3. How is the derivative of the Ad map calculated?

The derivative of the Ad map is calculated by taking the Lie bracket of the Lie algebra elements corresponding to the two elements being conjugated. This can also be expressed as the commutator of the left-invariant vector fields generated by the two elements.

4. Can the Ad map be extended to other types of groups?

Yes, the Ad map can be extended to other types of groups, such as Lie groups over other fields or topological groups. However, the definition and properties may differ slightly in these cases.

5. What are some applications of the derivative of the Ad map on a Lie group?

The derivative of the Ad map has various applications in differential geometry, Lie theory, and physics. It is used to define the Lie bracket on a Lie algebra, study the curvature of a Lie group, and understand the infinitesimal behavior of a group's action on a manifold, among others.

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