MIT 18.701 — Lecture 20

SUnSU_n and the Quaternions

§ Spheres and Balls

Definition (nn-Sphere). The nn-dimensional surface of a sphere is Sn:={x∈Rn+1:∣x∣=1}S^n := \{x \in \mathbb{R}^{n + 1} : |x| = 1 \}.

Definition (nn-Ball). The nn-dimensional interior of a sphere is Bn:={x∈Rn:∣x∣≤1}B^n := \{x \in \mathbb{R}^n : |x| \leq 1\}.

Example. The stereographic projection is a continuous, bijective map with continuous inverse (homeomorphism) from Sn∖{N}S^n \setminus\{N\} to Rn\mathbb{R}^n.

Analogously, reversing the stereographic projection yields a sense in which Rn∪{∞}≅Sn\mathbb{R}^n \cup \{\infty\} \cong S^n.

Example. Volumes behave very weirdly in higher dimensions!

For large nn, almost all of the volume of BnB^n comes from its outermost “shell”.

§ The Special Unitary Group SU2SU_2

Definition (Special Unitary Group). The special unitary group in Cn\mathbb{C}^n is:

SUn:=SLn(C)∩Un={A∈Cn×n∣AA†=In, det⁡A=1}.SU_n := SL_n(\mathbb{C}) \cap U_n = \{A \in \mathbb{C}^{n \times n} \mid AA^{\dagger} = I_n, \ \det A = 1\}.

Theorem. (SU2≅S3SU_2 \cong S^3) Matrices in SU2SU_2 naturally map bijectively to points on S3S^3.

Proof: If A=[αβγδ]A = \left[\begin{smallmatrix} \alpha & \beta \\ \gamma & \delta \end{smallmatrix}\right] is in SU2SU_2, then:

AA†=[∣α∣2+∣β∣2αγ‾+βδ‾α‾γ+β‾δ∣γ∣2+∣δ∣2]=[1001] and det⁡A=1   ⟺   δ=α‾ and γ=−β‾ and ∣α∣2+∣β∣2=1AA^{\dagger} = \begin{bmatrix} |\alpha|^2 + |\beta|^2 & \alpha \overline{\gamma} + \beta \overline{\delta} \\ \overline{\alpha}\gamma + \overline{\beta}\delta & |\gamma|^2 + |\delta|^2 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \text{ and } \det A = 1 ~ \iff ~ \delta = \overline{\alpha} \text{ and } \gamma = - \overline{\beta} \text{ and } |\alpha|^2 + |\beta|^2 = 1

Thus, every AA in SU2SU_2 is parametrized by (α,β)∈C2(\alpha, \beta) \in \mathbb{C}^2 with ∣α∣2+∣β∣2=1|\alpha|^2 + |\beta|^2 = 1.

If α=w+ix\alpha = w + ix and β=y+iz\beta = y + iz, we see SU2SU_2 is in fact {(w,x,y,z)∈R4∣w2+x2+y2+z2=1}\{(w, x, y, z) \in \mathbb{R}^4 \mid w^2 + x^2 + y^2 + z^2 = 1\}, which is S3S^3.   ■\blacksquare

Remark. Unfortunately, this is the nicest things will get.

§ The Quaternions H\mathbb{H}

We can rephrase our parametrization of SU2SU_2 as follows.

Theorem. (SU2SU_2 is a Span) Every matrix MM in SU2SU_2 is a linear combination in C2×2\mathbb{C}^{2 \times 2} of the form:

M=w[1001]+x[i00−i]+y[01−10]+z[0ii0],  given that w2+x2+y2+z2=1.M = w \left[\begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right] + x\left[\begin{smallmatrix} i & 0 \\ 0 & -i \end{smallmatrix}\right] + y\left[\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right] + z\left[\begin{smallmatrix} 0 & i \\ i & 0 \end{smallmatrix}\right], ~ \text{ given that } w^2 + x^2 + y^2 + z^2 = 1.

And here's the definition of the quaternions.

Definition (Quaternion). The quaternions are the set H=R+Ri+Rj+Rk\mathbb{H} = \mathbb{R} + \mathbb{R}i + \mathbb{R}j + \mathbb{R}k, constrained on:

i2=j2=k2=−1  and  ij=k=−ji  and  jk=i=−kj  and  ki=j=−iki^2 = j^2 = k^2 = -1 ~ \text{ and } ~ ij = k = -ji ~ \text{ and } ~ jk = i = -kj ~ \text{ and } ~ ki = j = -ik

(Those last three equations are analogous to cross products in R3\mathbb{R}^3.)

Definition (Length). The length of any α=w+xi+yj+zk∈H\alpha = w + xi + yj + zk \in \mathbb{H} is ∣α∣:=w2+x2+y2+z2|\alpha| := \sqrt{w^2 + x^2 + y^2 + z^2}.

Theorem. (SU2SU_2 are Unit Length H\mathbb{H}) Matrices in SU2SU_2 map bijectively with the elements w+xi+yj+zk∈Hw + xi + yj + zk \in \mathbb{H} with coefficients satisfying w2+x2+y2+z2=1w^2 + x^2 + y^2 + z^2 = 1.

Proof: The bijection f:{h∈H:∣h∣=1}→SU2f: \{h \in \mathbb{H} : |h| = 1\} \to SU_2 is via f:(i,j,k)↦([i00−i],[01−10],[0ii0])f: (i, j, k) \mapsto \left(\left[\begin{smallmatrix} i & 0 \\ 0 & -i \end{smallmatrix}\right], \left[\begin{smallmatrix} 0 & 1 \\ -1 & 0 \end{smallmatrix}\right], \left[\begin{smallmatrix} 0 & i \\ i & 0 \end{smallmatrix}\right] \right), which works.   ■\blacksquare

Remark. To really spell things out…

{r∈R:∣r∣=1}↔S0  and  {z∈C:∣z∣=1}↔S1  and  {h∈H:∣h∣=1}↔S3\{r \in \mathbb{R}: |r| = 1 \} \leftrightarrow S^0 ~ \text{ and } ~ \{z \in \mathbb{C} : |z| = 1 \} \leftrightarrow S^1 ~ \text{ and } ~ \{h \in \mathbb{H} : |h| = 1 \} \leftrightarrow S^3

Definition (Conjugation). Given an α=w+xi+yj+zk∈H\alpha = w + xi + yj + zk \in \mathbb{H}, its conjugate is α‾:=w−xi−yj−zk\overline{\alpha} := w - xi - yj - zk.

Theorem. (Conjugation Properties) For any α,β∈H\alpha, \beta \in \mathbb{H}, we have:

α+β‾=α‾+β‾   and   α⋅β‾=β‾⋅α‾   and   α⋅α‾=∣α∣2   and   ∣α⋅β∣=∣α∣⋅∣β∣\overline{\alpha + \beta} = \overline{\alpha} + \overline{\beta} ~~ \text{ and } ~~ \overline{\alpha \cdot \beta} = \overline{\beta} \cdot \overline{\alpha} ~~ \text{ and } ~~ \alpha \cdot \overline{\alpha} = |\alpha|^2 ~~ \text{ and } ~~ |\alpha \cdot \beta| = |\alpha| \cdot |\beta|

Proof: Just plain computation. It's worth seeing how the fourth identity follows from the second and third!   ■\blacksquare

§ Sidenote: Linear Algebra Over H\mathbb{H}

Definition (Skew Field). A skew field (also called a division algebra, or a corps non commutatif in French) is a field with not-necessarily-commutative multiplication.

Theorem. (Skew Fields Over R\mathbb{R}) The only skew fields with finite dimension as R\mathbb{R}-vector spaces are R\mathbb{R}, C\mathbb{C}, and H\mathbb{H}.

It also turns out that you can do almost all of linear algebra over skew fields!

Definition (Vector Space Over Skew Fields). Let FF be a skew field. Then VV is a vector space over FF if it has commutative addition, right scalar multiplication, and left matrix multiplication.

Under this definition, things like GLn(H)GL_n(\mathbb{H}) still make sense! Indeed, GLn(H)GL_n(\mathbb{H}) is just a subgroup of GL4n(R)GL_{4n}(\mathbb{R}).

Theorem. (Classification of Compact Connected Lie Groups) Suppose GG is a connected, bounded, and closed (under limits) subgroup of GLn(R)GL_n(\mathbb{R}). Then it must be that G≅(∏iHi)/KG \cong (\prod_i H_i) / K, where:

§ Sidenote: A Number Theoretic Application

Recall that the identity ∣u⋅v∣2=∣u∣2⋅∣v∣2|u \cdot v|^2 = |u|^2 \cdot |v|^2 for u,v∈Cu, v \in \mathbb{C}, upon taking u=a+biu = a + bi and v=c+div = c + di, yields:

(a2+b2)⋅(c2+d2)=(ac−bd)2+(ad+bc)2.(a^2 + b^2) \cdot (c^2 + d^2) = (ac - bd)^2 + (ad + bc)^2.

This gives you the following theorem:

Theorem. (Sums of Two Squares) Let S⊆Z+S \subseteq \mathbb{Z}^+ be the set of all positive integers that are the sums of two squares. Then SS is multiplicatively closed.

The quaternionic version of this analogously gives another theorem:

Theorem. (Sums of Four Squares) Let S⊆Z+S \subseteq \mathbb{Z}^+ be the set of all positive integers that are the sums of four squares. Then SS is multiplicatively closed.

(In fact, in the above theorem, it turns out that S=Z+S = \mathbb{Z}^+.)

However, if we replace “four squares” with “three squares”, it turns out this theorem is NOT true! For example, 3=12+12+123 = 1^2 + 1^2 + 1^2 and 5=22+12+025 = 2^2 + 1^2 + 0^2, but 1515 is not the sum of three squares.