User talk:Fropuff/Drafts/n-sphere

Here is some material I wrote in Latex. Feel free to canabilize it anyway you want. If you don't think any of it is relevent to your development, feel free to ignore it.

Recall the quaternions $\mathbb{H}=\{a+x\mathbf{i}+y\bj +z\bk|a,x,y,z \in \mathbb{R}\}$. More conveniently we can write a quaternion as the sum of a real number and a vector in $\mathbb{R}^3$, for instance $q=a+\overline{v}$, $q'=b+\ow$. The product of two quaternions is given in terms of operations between vectors and scalars, \[qq'= (a+\ov)(b+\ow)=ab-\ov.\ow+a\ow+b\ov+\ov\times\ow.\]

The conjugate of a quaternion is given by changing the sign of its vector part, \[ \overline{q}=a-\ov.\] The norm squared is \[ ||q||^2=q\overline{q}=a^2+\ov.\ov.\] Finally, the unit quaternions are the quaternions whose whose norm square is $1$. They form a group which is called $SU(2)$. The identity is $1$. The tangent space of $SU(2)$ at $1$ consists of all purely imaginary quaternions, that is \[ su(2)=\{x\bi+y\bj+z\bk\}=\mathbb{R}^3.\] The tangent space at $q$, $T_qSU(2)$, is $su(2)q$. There is a convenient way of producing a path in $SU(2)$ through $1$ having a tangent vector $\ov$. Recall that if $\ov$ and $\ow$ point in the same direction then $\ov \times \ow=0$. The product of two elements of $SU(2)$ whose vector part has the same direction is a third quaternion whose vector part has the same direction. Hence exponential map $\mathrm{exp}:\mathbb{R}^3 \rightarrow SU(2)$ can be written

\[ \mathrm{exp}(\ov)=\sum_{n=0}^{\infty} \frac{\ov^n}{n!}=\cos{(||v||)}+\sin{(||v||)}\frac{\ov}{||\ov||}.\]

Let $\mathrm{tr}:SU(2) \rightarrow [-2,2]$ be the map that sends $a+\ov$ to $2a$. Notice that $\mathrm{tr}(\mathrm{exp}(\ov)=2\cos{(||\ov||)}$ One can easily compute $\tr(q_1q_2)=\tr(q_2q_1)$.

The purely imaginary quaternions, $\mathbb{R}^3$, are a Lie algebra by virtue of being the tangent space to a Lie group at the identity. The Lie bracket is given by twice the crossproduct, that is $[\ov,\ow]=2\ov \times \ow$.

Notice that $SU(2)$ acts on $\mathbb{H}$ by conjugation, that is $ q.h=qh\overline{q}$. This action preserves the purely imaginary quaternions and gives rise to an action of $SU(2)$ on $su(2)$ which we call the adjoint action. The adjoint action has kernel $\{1,-1\}$ and so descends to an action of $PU(2)=SU(2)/\{1,-1\}$. As the adjoint action preserves the inner product on $su(2)=\mathbb{R}^3$, it can be seen to factor through $SO(3)$. In fact, $PU(2)=SO(3)$. This means that the standard inner product on $\mathbb{R}^3$ is preserved by the adjoint action of $SU(2)$. If $\rho: G \rightarrow SU(2)$ is a homomorphism, we get an action on $su(2)$ by composing with the adjoint action. We denote this by $\ad\rho$.

A word map is a map $w:SU(2)^n \rightarrow SU(2)$ that sends the tuple $(A_1,\ldots A_n)$ to the element of $SU(2)$ obtained by instantiating a word in the variables $a_i$ with the  matrices $A_i$. A word map is differentiable: using the identifications of tangent spaces with $su(2)$ via right translation defines \[ dw_{(A_1,\ldots A_n)}:\oplus_n su(2) \rightarrow su(2).\] Let $\frac{\partial w}{\partial a_i}$ be the Fox free partial derivatives of the word $w$ with respect to the variables $a_i$ instantiated with the $A_i$. Then the derivative is \[ dw_{(A_1,\ldots A_n)}(X_1,\ldots X_n)=\sum_i \frac{\partial w}{\partial a_i}.X_i\] where the period means we are using the adjoint action.

There is one irreducible representation of $SU(2)$ $\underline{m}$ for each nonnegative integer $0$. The dimension of $\underline{m}$ is $m+1$. We denote the trace in this representation by $\tr_m$. The heat kernel trace is the function \[ \sum_{c=0}^{\infty} exp(-c(c+2)/\lambda)(c+1)\tr_c.\] It is smooth, positive, its integral against Haar measure is one, and as $\lambda\rightarrow \infty$ it gets smaller and smaller at any point in $SU(2)$ other than the identity.

Haar measure on $SU(2)$ is obtained from the standard volume form for $SU(2)\subset \mathbb{R}^3$ by dividing by $2\pi^2$, so that the total volume is equal to $1$. We will generally deal with Haar measure by using the trace function $\mathrm{tr}:SU(2) \rightarrow [-2,2]$ to push forward to an interval. The push-forward measure is given by $\frac{1}{2\pi}\sqrt{4-x^2}\mathrm{dx}$. Under this map the trace is sent to $x$. The trace in the irreducible representation $\underline{m}$ is given by $s_m$, where $s_0=1$, $s_1=x$ and $s_m=xs_{m-1}-s_{m-2}$. We can expand the $s_m$ as Taylor series at $x=2$ to get, \[ s_m=\sum_{i=0}^m \binom{m+i+1}{2i+1}(x-2)^i.\]