SO(3) as a quotient group of SU(2)?

In summary, there is a two to one homomorphism from SU(2) to SO(3), and SO(3) can be interpreted as a quotient group of SU(2) with respect to the subgroup {I,-I}. This is because {I,-I} is an abelian normal subgroup of SU(2), and quotient groups are always trivially related to a group and its normal subgroup. Additionally, both SU(2) and SO(3) can be regarded as 3-spheres, and after dividing out by {I,-I}, the quotient group SU(2)/{I,-I} is isomorphic to SO(3).
  • #1
wdlang
307
0
we know there is a two to one homomorphism from SU(2) to SO(3)

suppose u is an element in SU(2)

then u and -u map into the same element in SO(3)

the question is, maybe SO(3) is a quotient group of SU(2)? with respect to the subgroup {I,-I}?
 
Physics news on Phys.org
  • #2
Yes, SO(3) can be interpreted as a quotient group of SU(2).

{I,-I} is an abelian subgroup and it is a normal subgroup of SU(2). A quotient group G/H is always trivially related to a group G and a normal subgroup H of G.

Thus, indeed SO(3) equals (or is at least isomorphic to) SU(2) after dividing out by {I,-I}.

Another way to see it:

The 3-sphere (S^3) can be considered as a 'manifold' isomorphic to SU(2) and the real projective space of dimension 3, which is a quotient of S^3 under 'antipodal identification' is isomorphic to SO(3). Both are orientable manifolds.
 
Last edited:
  • #3
{I,-I} is an abelian subgroup and it is a normal subgroup of SU(2). A quotient group G/H is always trivially related to a group G and a normal subgroup H of G.

Thus, indeed SO(3) equals (or is at least isomorphic to) SU(2) after dividing out by {I,-I}.
---------------------------------------------------------------------------------------------------
yes, you have shown that SU(2)/{I,-I} is a quotient group

but i think you did not show that it is isomorphic to SO(3).
 
  • #4
Well, you didn't ask to show it.

I wrote: "The 3-sphere (S^3) can be considered as a 'manifold' isomorphic to SU(2) and the real projective space of dimension 3, which is a quotient of S^3 under 'antipodal identification' is isomorphic to SO(3). Both are orientable manifolds."

Working out the defining relations of any matrix in the fundamental representation of SU(2) will give us that SU(2) can be regarded as a 3-sphere. Doing the same for SO(3) also gives us that SO(3) can be regarded as a 3-sphere, however, an antipodal pair of points will give us the same element of SO(3).

Another way to see it: The adjoint representation of SU(2) can be identified with the fundamental representation of SO(3).
 
  • #5


I can confirm that the statement is correct. The special unitary group SU(2) is a double cover of the special orthogonal group SO(3). This means that for every element in SO(3), there are two corresponding elements in SU(2) that map to it. This is due to the fact that SU(2) has a two-to-one homomorphism onto SO(3). Therefore, SO(3) can be seen as a quotient group of SU(2) with respect to the subgroup {I,-I}. This relationship between the two groups is important in various areas of physics, such as quantum mechanics and particle physics.
 

1. What is the relationship between SO(3) and SU(2)?

SO(3) is a subgroup of SU(2), meaning that it is a smaller group that is contained within the larger group SU(2). This means that all elements of SO(3) are also elements of SU(2), but not all elements of SU(2) are elements of SO(3).

2. How does SO(3) act as a quotient group of SU(2)?

SO(3) acts as a quotient group of SU(2) by taking the elements of SU(2) that are not in SO(3) and grouping them together in a single coset. This creates a new group, where each element of SO(3) is equivalent to the identity element of the new group.

3. What is the significance of SO(3) being a quotient group of SU(2)?

The significance of SO(3) being a quotient group of SU(2) lies in its applications in physics and mathematics. In physics, SO(3) is used to describe the rotational symmetry of physical systems, while SU(2) is used to describe the properties of quantum particles. The relationship between the two groups allows for a deeper understanding of these concepts.

4. How are the elements of SO(3) and SU(2) related to each other?

The elements of SO(3) and SU(2) are related through a mathematical mapping known as the spinor map. This map takes an element of SU(2) and maps it to an element of SO(3). This relationship is important in understanding the properties of both groups.

5. What are some real-world examples of SO(3) as a quotient group of SU(2)?

One real-world example of SO(3) as a quotient group of SU(2) is in the study of molecular symmetry. The rotational symmetry of molecules can be described using SO(3), while the properties of the atoms within the molecule can be described using SU(2). Another example is in the study of elementary particles, where SU(2) is used to describe the properties of particles and SO(3) is used to describe their rotational symmetry.

Similar threads

  • Topology and Analysis
Replies
16
Views
453
  • Topology and Analysis
2
Replies
61
Views
884
  • Differential Geometry
Replies
11
Views
672
  • Linear and Abstract Algebra
Replies
1
Views
430
  • Differential Geometry
Replies
3
Views
1K
Replies
7
Views
739
  • Linear and Abstract Algebra
Replies
1
Views
877
  • High Energy, Nuclear, Particle Physics
Replies
10
Views
463
  • High Energy, Nuclear, Particle Physics
Replies
2
Views
494
  • Classical Physics
Replies
6
Views
462
Back
Top