Simple question about SU(2) and SO(3)

In summary: The center of SU(2) maps to the identity of SO(3). Thus, a path in SU(2) from the identity to the other element in the center projects onto a loop in SO(3). This loop is non-trivial, and is a representative of the single non-trivial homotopy class of loops in SO(3). This is a general thing, as the fundamental group of a simply connected space G and a surjective map of it onto another space H is isomorphic to the kernel of the map. Therefore, the kernel of the surjective homomorphism from SU(2) to SO(3) is isomorphic to the fundamental group of SO(3), which is Z/2Z.
  • #1
Jim Kata
197
6
there's a surjective homomorphism from

a : SU(2) --> SO(3)

The kernel of this homomorphism is the center of SU(2) which is Z/2Z. Now the fundamental group of SO(3) is Z/2Z. This is a general thing.

The simplest version of my question is how is the center of SU(2) related to the fundamental group SO(3).
 
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  • #2
Consider a path in SU(2) running from the identity to the other element in the center. Since SU(2) is simply connected, there is essentially one such path (ie, up to homotopy). Now project this path onto SO(3) using the surjective homomorphism. Since the center of SU(2) gets mapped to the identity of SO(3), this projected path starts and ends at the identity, so is a loop, and so defines an element of the fundamental group of SO(3).

To see this is a non-trivial loop, ie, can't be continuously shrunk to the constant loop, note that if we deform this loop slightly, we can mirror this deformation of the original path in SU(2) so that our new loop is the projection of this new path. Now, if it was possible to iterate this procedure until our path was shrunk to the constant loop, the corresponding path in SU(2) would have to have shrunk to the constant path as well. But remember the loop in SO(3) always has both ends at the identity, so the corresponding path in SU(2) must always have each of its endpoints at one of the two points in the center. But it's clearly impossible that we could continuously deform a path connecting two distinct points to a loop ending at a single one of the points through a series of intermediate paths always ending at the two points: there would have to be a discontinuous jump at some point. Thus this is impossible, and the loop is a representative of the single non-trivial homotopy class of loops in SO(3).

In general, if we have a simply connected space G and a surjective map of it onto another space H, the elements of the fundamental group of H are in one to one correspondence with the sheets of this covering (ie, with the elements in the preimage of a single point). In the case where G and H are groups and the map is a homomorphism, the fundamental group is isomorphic to the kernel of the map.
 
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  • #3
Thank you, that was a clear explanation. Sorry, I'm not a math guy, and prefer to just ask these questions instead of really thinking about it. I guess a follow up question would be say you look at the complexifications of the groups, is the same statement is true? The center of Sl(2,C) is isomorphic to the fundamental group of PSL(2,C) under this morphism. I think that's true, but can it be extended?

Given a surjective homorphism from a simply connected group to some other group and taking the complexifications of both the fiber group and the target group is the statement still true? That fundamental group of the complexified group in the image of the homomorphism is the same as the kernel of the homomorphism from the complexified group in the pre-image.
 
  • #4
StatusX said:
Consider a path in SU(2) running from the identity to the other element in the center. Since SU(2) is simply connected, there is essentially one such path (ie, up to homotopy). Now project this path onto SO(3) using the surjective homomorphism. Since the center of SU(2) gets mapped to the identity of SO(3), this projected path starts and ends at the identity, so is a loop, and so defines an element of the fundamental group of SO(3).

To see this is a non-trivial loop, ie, can't be continuously shrunk to the constant loop, note that if we deform this loop slightly, we can mirror this deformation of the original path in SU(2) so that our new loop is the projection of this new path. Now, if it was possible to iterate this procedure until our path was shrunk to the constant loop, the corresponding path in SU(2) would have to have shrunk to the constant path as well. But remember the loop in SO(3) always has both ends at the identity, so the corresponding path in SU(2) must always have each of its endpoints at one of the two points in the center. But it's clearly impossible that we could continuously deform a path connecting two distinct points to a loop ending at a single one of the points through a series of intermediate paths always ending at the two points: there would have to be a discontinuous jump at some point. Thus this is impossible, and the loop is a representative of the single non-trivial homotopy class of loops in SO(3).

In general, if we have a simply connected space G and a surjective map of it onto another space H, the elements of the fundamental group of H are in one to one correspondence with the sheets of this covering (ie, with the elements in the preimage of a single point). In the case where G and H are groups and the map is a homomorphism, the fundamental group is isomorphic to the kernel of the map.

well done explanation
 
  • #5
Jim Kata said:
there's a surjective homomorphism from

a : SU(2) --> SO(3)

The kernel of this homomorphism is the center of SU(2) which is Z/2Z. Now the fundamental group of SO(3) is Z/2Z. This is a general thing.

The simplest version of my question is how is the center of SU(2) related to the fundamental group SO(3).

SU(2) is simply connected - the homomorphism, SU(2) --> SO(3), is a covering projection.
 

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

SU(2) and SO(3) are both mathematical groups used to describe the symmetries of physical systems. The main difference between them is that SU(2) is a complex group, meaning it contains elements with both real and imaginary components, while SO(3) is a real group, containing only real elements. In terms of their applications, SU(2) is often used in quantum mechanics and particle physics, while SO(3) is used in classical mechanics and rotational symmetry.

2. How are SU(2) and SO(3) related?

SU(2) and SO(3) are closely related, as they both describe the symmetries of three-dimensional space. In fact, SU(2) is the double cover of SO(3), meaning that every element in SO(3) has two corresponding elements in SU(2). This relationship is important in understanding the spin of particles in quantum mechanics, as spin is described by SU(2) representations.

3. What does the "U" in SU(2) stand for?

The "U" in SU(2) stands for "unitary", which is a mathematical property of complex matrices. Unitary matrices have the property that their conjugate transpose is equal to their inverse. In the context of SU(2), this means that the group's elements preserve the length of vectors in complex vector spaces.

4. How are SU(2) and SO(3) used in physics?

SU(2) and SO(3) are used in physics to describe the symmetries of physical systems. In particular, SU(2) is used to describe the symmetries of spin in quantum mechanics, while SO(3) is used to describe rotational symmetry in classical mechanics. These groups are also used in particle physics and cosmology, as they play important roles in understanding the fundamental laws of the universe.

5. Are there any other related mathematical groups to SU(2) and SO(3)?

Yes, there are other related mathematical groups to SU(2) and SO(3). For example, SU(3) is the complex group used to describe the symmetries of strong interactions in particle physics, while SO(4) is used to describe the symmetries of four-dimensional space. In general, groups such as SU(N) and SO(N) can be used to describe the symmetries of N-dimensional space.

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