Does SU(2) Commutativity Conflict with Its Non-Euclidean Structure?

[Kontilera]

Hello!

I thought I had a good picture of this stuff but now I suspect that I've mixed up things completely..
For su(2) we have:
$$f_{ab}\,^c = \epsilon_{ab}\,^c.$$
Which means that, for the basis of our tangentspace we get:
$$[\partial_a, \partial_b] = i \epsilon_{ab}\,^c\, \partial_c$$
Where we can see that the space of linear approximations is closed under the Lie bracket product.

But what about the definition of a smooth manifold being locally euclidean?
For R^n we know that our orthogonal partial derivatives commute,
$$\partial_x\partial_y - \partial_y\partial_x = 0.$$

Shouldnt we then expect the orthogonal derivatives to commute in su(2) as well?
This would mean that our structure constants are 0 both when a=b and when a is not equalt to b (for a orthonomal basis).
In other words.. it looks like the Lie algebra should be quite trivial.

This is of course not the case, but where does my mind make the mistake in this resoning?

Thanks!
// Kontilera

 

[fresh_42]

The Lie algebra elements are tangents along a path in the group. That means at its endpoint we are in a different tangent space albeit still isomorphic to $\mathbb{R}^n$. But $\partial_a\, , \,\partial_b$ have three tangent spaces they arose from: starting point for $a$, endpoint = starting point for $b$, and endpoint of $b$. The commutator measures how the endpoints of this path $b\circ a$ differs from that of $a \circ a$. Since we have different tangent spaces at different points, we cannot expect that all this happens within one tangent space of the group. This is the difference between local and global behavior. Within a single $\mathbb{R}^n$ the paths $b\circ a$ and $a\circ b$ lead to the same endpoint, with changing tangent spaces this is no longer guaranteed.

Have a look on these insights:
https://www.physicsforums.com/insights/pantheon-derivatives-part-iv/https://www.physicsforums.com/insights/journey-manifold-su2-part-ii/

 

[math771]

Hi Kontilera,

It seems like you are trying to understand the relationship between the Lie algebra su(2) and the concept of a smooth manifold being locally euclidean. The Lie algebra su(2) has a non-trivial structure, as you mentioned, and this can be seen from the fact that the structure constants are not all zero. This means that the Lie bracket product of two tangent vectors in su(2) is not always zero, and therefore the space of linear approximations is not closed under the Lie bracket product.

As for the definition of a smooth manifold being locally euclidean, this refers to the fact that a smooth manifold can be locally approximated by an euclidean space. This does not necessarily mean that all the properties of euclidean spaces also hold for smooth manifolds. In particular, the commutativity of orthogonal partial derivatives is a property of euclidean spaces, but it does not necessarily hold for all smooth manifolds.

In the case of su(2), the orthogonal derivatives do not commute, as you correctly pointed out. This does not mean that there is a mistake in your reasoning, but rather that the concept of smooth manifolds being locally euclidean does not directly apply to su(2). The structure of su(2) is more complex, and it cannot be fully captured by the concept of a smooth manifold being locally euclidean.

I hope this helps clarify your understanding. Keep exploring and asking questions!