Properties and Elements of SO(4) Group in 4 Dimensions

[jobinjosen]

What are the properties of SO(4) group? , How this acts as a rotator in 4 dimensions?, What are the elements of Rotation matrix in a specific dimension among four dimensions?

 

[jostpuur]

I've been waiting for some kind of answer for this post too. I cannot answer the OP, but I'll throw more questions

When a rotation is carried out in three dimensions, there is an axis of rotation, that is a one dimensional subspace of the three dimensional space, and the rotation is in fact just a two dimensional rotation in the orthogonal complement of this axis. In analogy with this I might guess, that in four dimensions the one dimensional axis is replaced by a two dimensional subspace, that is then some kind of "axis" of rotation. Is this correct?

In analogy with SO(3), I might guess that $SO(4)=\textrm{exp}(\mathfrak{so}(4))$, where $\mathfrak{so}(4)$ consists of those 4x4 matrices that are antisymmetric (satisty $X^T=-X$). However, these matrices depend only on 6 real variables, which is not enough to define two four dimensional vectors that would span the "axis space", so it seems I'm guessing something wrong.

 

[George Jones]

In analogy with SO(3), I might guess that $SO(4)=\textrm{exp}(\mathfrak{so}(4))$, where $\mathfrak{so}(4)$ consists of those 4x4 matrices that are antisymmetric (satisty $X^T=-X$).

This is true of SO(n) and so(n).

However, these matrices depend only on 6 real variables, which is not enough to define two four dimensional vectors that would span the "axis space", so it seems I'm guessing something wrong.

https://www.physicsforums.com/showpost.php?p=1110359&postcount=20 may be of interest to both you and jobinjosen.

 

[jobinjosen]

Here are some more points regarding SO(4) group.

In SO(3) rotations, generator of rotation are components of Angular momentum (Lx, Ly, Lz) for rotation w.r.t corresponding axis.

Now, In SO(4), what are the generators of rotation?

They are components of Angular momentum (Lx, Ly, Lz) and components of Laplace Runge Lenz (LRL) vector (Ax, Ay, Az). Constancy of this LRL vector creates aditional symmetry. Am I correct?