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The Identity of SU(2) and S3

Thomas J. McFarlane
13 March 1992
revised 26 November 2004

Introduction

From superficial equations of arithmetic to the deepest theorems, mathematics is everywhere showing us the identity behind apparent differences. As a particular instance of this phenomenon, this article will show the sense in which SU(2) and S3 may be regarded as the same. In particular, we will show that they are diffeomorphic as smooth manifolds. Moreover, they are isomorphic as Lie groups, where we give S3 algebraic structure through identification with the group of unit quaternions.

Background

In order to demonstrate the sense in which S3 and SU(2) are the same, we must first understand them as distinct objects, and then understand what it means to call them equivalent. To this end, we define smooth manifolds and Lie groups, and then diffeomorphisms and Lie isomorphisms.

Definition A manifold is a locally Euclidean Hausdorff space with a countable basis of open sets.

Definition A smooth manifold is a manifold together with a smooth structure. (Roughly speaking, a smooth structure defines local coordinates on the manifold that vary smoothly.)

In particular, Rn is a smooth manifold. And it can be shown that both

S3 = { (x1,x2,x3,x4) ∈ R4 : x12+x22+x32+x42=1 }

and

SU(2) = { A ∈ Mat(2,C) : A*A=I and det{A}=1 }

are smooth manifolds. Note: Mat(2,C) is the set of 2×2 matrices with complex-valued entries.

Definition A diffeomorphism between two manifolds is a smooth homeomorphism whose inverse is also smooth. We say two manifolds are diffeomorphic when there is a diffeomorphism between them.

These definitions give precise meaning to one sense in which S3 and SU(2) are the same, namely, they are diffeomorphic as smooth manifolds. To give precise meaning to the second sense in which they are the same we require two more definitions.

Definition A Lie group is a smooth manifold that is also an algebraic group whose operations (composition and inversion) are smooth maps. (A Lie group is a special case of a topological group since smooth maps are a special case of continuous maps.)

Definition A Lie isomorphism between Lie groups is a diffeomorphism that is also a group homomorphism (and hence group isomorphism). We say two Lie groups are (Lie) isomorphic if there is a Lie isomorphism between them.

The group SU(2) is a Lie subgroup of the general linear group GL(2,C). The manifold S3, on the other hand, has no inherent group structure. But we can give it a group structure by identifying it with the unit quaternions

SH = { a+bi+cj+dk ∈ H : a2+b2+c2+d2=1 },

where the group operation is quaternion multiplication (determined by i2=j2=k2=ijk=-1) and the group inverse is quaternion conjugation (defined by γ*=a-bi-cj-dk where γ=a+bi+cj+dk ∈ H) since γγ*=1 for all γ ∈ SH. Since these operations are smooth, SH is a Lie group. In particular, it is a Lie subgroup of the quaternions H.

With this identification of S3 with SH we can now give precise meaning to the second sense in which S3 and SU(2) are the same, namely, SH and SU(2) are isomorphic as Lie groups. To prove this, we first construct a map φ : H → Mat(2,C) as follows.

First note that there is an injection C → H given by a+ib → a+bi and a bijection C2 ↔ H via the correspondence
(
a+ib
c+id
) ↔ a+bi+cj-dk = (a+bi)+j(c+di),
or, more succinctly,
(
z1
z2
) ↔ z1+jz2,
where we identify i ∈ C with i ∈ H.

The above correspondence C2↔H allows us to use quaternion multiplication in H to define a linear transformation of vectors in C2. First, observe that an element γ=w1+jw2 ∈ H defines a transformation γ : H → H given by z1+jz2 → γ⋅(z1+jz2). Expanding this product, using the fact that wj=jw*, one finds that γ⋅(z1+jz2) = (w1+jw2)⋅(z1+jz2) = w1z1+jw2jz2+jw2z1+w1jz2 = (w1z1-w2*z2)+j(w2z1+w1*z2). Thus, the quaternion multiplication γ : H → H corresponds to a linear transformation φ(γ) : C2 → C2 given by
(
z1
z2
) → (
w1z1-w2*z2
w2z1+w1*z2
).
In other words,
φ(γ) (
z1
z2
) = (
w1z1-w2*z2
w2z1+w1*z2
) = (
w1 -w2*
w2 w1*
) (
z1
z2
).
Thus,
φ(γ) = (
w1 -w2*
w2 w1*
).
This gives us a map φ : H → Mat(2,C) given by γ=w1+jw2 → φ(γ). Now we are ready to show that SH is Lie isomorphic to SU(2).

Theorem
SU(2) is diffeomorphic to S3 and Lie-isomorphic to SH.

Proof
First assume γ ∈ SH. Then we can easily verify that det(φ(γ)) = w1*w1+w2*w2 = 1 and φ(γ)*φ(γ) = I . So φ(γ) ∈ SU(2). Thus, φ : SH → SU(2).

Conversely, let A ∈ SU(2). Then the constraints det(A)=1 and A*A=I together imply that A must have the form
A = (
w1 -w2*
w2 w1*
),
for some w1,w2 ∈ C. Now if we select γ=φ-1(A)=w1+jw2 ∈ H, then we have a map φ-1 : SU(2) → H, and we see that φ(γ)=A. Moreover, since γ*&gamma = w1*w1+w2*w2 = 1, it follows that γ ∈ SH. Therefore, φ : SH → SU(2) is a bijection.

From the above discussion it is clear that φ and φ-1 are smooth maps since their components are linear functions of the coordinates. Moreover, since SU(2) and SH are submanifolds, the restrictions of φ and φ-1 to these submanifolds are also smooth. Therefore, φ is a diffeomorphism, and so S3=SH is diffeomorphic to SU(2).

Finally, we show that φ is also a Lie isomorphism between SH and SU(2). First, observe that φ is a group homomorphism. Indeed, φ(γ*) = φ((w1+jw2)*) = φ(w1*-jw2) = φ(γ)* = φ(γ)-1, and (γ1γ2)(z1+jz2) = (γ1)(γ2)(z1+jz2) implies that φ(γ1γ2) = φ(γ1)φ(γ2). Therefore, φ is a Lie homomorphism. And since φ is a diffeomorphism, φ is a Lie isomorphism. This completes the proof. ♦

Conclusion

From a superficial glance at the definitions for SU(2) and S3, one would not expect that they are diffeomorphic manifolds. Nor would one guess that SU(2) and SH are isomorphic Lie groups. Yet, with the insights afforded by mathematics we have revealed the underlying structural identity between these two objects.