Andrius: I'm investigating
The Mechanism of Bott Periodicity
Eighth roots of unity
The eighth root of unity {$e^{\frac{\pi}{2}i}=\frac{\sqrt{2}}{2}(1+i)$} has period {$8$}.
Notably, powers of {$e^{\frac{\pi}{2}i}=1+i$} brings to mind the Clifford algebra representations, their break down and their powers.
- 1
- 1+i
- 2i
- -2+2i
- -4
- -4-4i
- -8i
- 8-8i
- 16
Note that if we ignore the magnitude, and simply focus on the angle, on the orientation, then we get back to where we started.
Clifford algebra representations
The Clifford algebra representations can be calculated from the following intial conditions and recursive relations. See especially: José Figueroa-O’Farrill. Spin Geometry.
The initial conditions are that
| {$M_2(\mathbb{R})=Cl(1,1)$} | ||
| {$\mathbb{R}\oplus\mathbb{R}=Cl(1,0)$} | {$M_2(\mathbb{R})=Cl(1,1)$} | |
| {$\mathbb{R}=Cl(0,0)$} | {$\mathbb{C}=Cl(0,1)$} | {$\mathbb{H}=Cl(0,2)$} |
The recursion relations are
{$Cl(0,n+2)\cong Cl(n,0)\otimes Cl(0,2)$}
{$Cl(n+2,0)\cong Cl(0,n)\otimes Cl(2,0)$}
In the positive definite direction, the initial conditions are {$\mathbb{R}$} and {$\mathbb{R}\oplus\mathbb{R}$} and then {$M_2(\mathbb{R})$} and {$M_2(\mathbb{C})$} which apply {$M_2(\mathbb{R})$} to {$\mathbb{R}$} and {$\mathbb{C}$}. Then they are tensored repeatedly by {$M_2(\mathbb{H})$}.
In the negative definite direction, the initial conditions are {$\mathbb{R}$} and {$\mathbb{C}$} and then {$\mathbb{H}$} and {$\mathbb{H}\oplus\mathbb{H}$} which apply {$\mathbb{H}$} to {$\mathbb{R}$} and {$\mathbb{R}\oplus\mathbb{R}$}. Then they are tensored repeatedly by {$M_2(\mathbb{H})$}.
By analogy, we have
| {$M_2(\mathbb{R})\leftrightarrow +i$} | {$\mathbb{H}\leftrightarrow -i $} | {$M_2(\mathbb{H})\leftrightarrow -1$} | {$M_2(\mathbb{H})\otimes M_2(\mathbb{H})\cong M_4(\mathbb{H})\leftrightarrow +1$} |
The structure {$M_2(\mathbb{H})\otimes M_2(\mathbb{H})$}, which places a {$2\times 2$} matrix of quaternions aside each entry of another such {$2\times 2$} matrix of quaternions, can be thought of as describing the pre-frame and post-frame of a division of everything, which in this case is simply the nullsome. Thus a frame is defined by the foursome {$M_2(\mathbb{H})$} where the explicit structure is given by the {$2\times 2$} choices {$M_2(\mathbb{R})$} and the implicit structure is given by {$H$} which is a complex number of a complex number, or alternatively, a three-cycle with an extra dimension.
Compare this with the case of complex Clifford algebras. The initial conditions are {$\mathbb{C}$} and {$\mathbb{C}\oplus\mathbb{C}$} and then we repeatedly tensor by {$M_2(\mathbb{C})$}.
We can study this as a manifestation of choice as given by {$J_1=i$} so that we have {$(1+J_1)=(1+i)$}.
Spin representations
Spin representations can be understood as restrictions of Clifford algebra representations.
Bilinear forms
There are eight bilinear forms {$\epsilon$}.
- {$\mathbb{R}$}-symmetric is {$\mathbb{R}$}-bilinear with {$\epsilon(x,y)=\epsilon(y,x)$}.
- {$\mathbb{R}$}-skew is {$\mathbb{R}$}-bilinear with {$\epsilon(x,y)=-\epsilon(y,x)$}.
- {$\mathbb{C}$}-symmetric is {$\mathbb{C}$}-bilinear with {$\epsilon(x,y)=\epsilon(y,x)$}.
- {$\mathbb{C}$}-skew is {$\mathbb{C}$}-bilinear with {$\epsilon(x,y)=-\epsilon(y,x)$}.
- {$\mathbb{C}$}-hermitian symmetric is {$\mathbb{C}$}-bilinear with {$\epsilon(x,y)=\overline{\epsilon(y,x)}$}.
- {$\mathbb{C}$}-hermitian skew is {$\mathbb{C}$}-bilinear with {$\epsilon(x,y)=-\overline{\epsilon(y,x)}$}.
- {$\mathbb{H}$}-hermitian symmetric is {$\mathbb{H}$}-bilinear with {$\epsilon(x,y)=\overline{\epsilon(y,x)}$}.
- {$\mathbb{H}$}-hermitian skew is {$\mathbb{H}$}-bilinear with {$\epsilon(x,y)=-\overline{\epsilon(y,x)}$}.
There are seven inner products.
Adjoints: Antiautomorphisms
For each of the seven types of inner products there is an antiautomorphism called the adjoint.
{$\epsilon(ax,y)=\epsilon(x,a*y)$} for all {$x,y\in V$}
Super division algebras
This relates the three division algebras {$\mathbb{R},\mathbb{C},\mathbb{H}$} with the seven super division algebras.
The even part {$A$} of an associative superalgebra must be either {$\mathbb{R}, \mathbb{C}$} or {$\mathbb{H}$}.
The even part {$A$} may be all of the superalgebra, as in the case of the division algebras. Otherwise, the odd part is isomorphic to the even part. And multiplying by a generator {$e$} manifests that isomorphism. In each case, {$\langle e, e \rangle$} must be real.
For any {$a\in A$} and generator {$e$} there exists a unique {$a'\in A$} such that {$ae=ea'$}.
- There is one {$\mathbb{R}$}-automorphism and we can set {$e^2=\pm 1$}.
- There are two {$\mathbb{C}$}-automorphisms, the identity (in which case we can set {$e^2=1$} and we are in complex Bott periodicity) and conjugation (in which case we can set {$e^2=\pm 1$}.
- There is one {$\mathbb{H}$}-automorphism and we can set {$e^2=\pm 1$}.
Real Bott periodicity takes us between the division algebras {$\mathbb{R}$} and {$\mathbb{H}$}. Complex Bott periodicity takes us between the division algebra {$\mathbb{C}$} and itself.
Note that {$\mathbb{H}\otimes M_2(\mathbb{R})$} can be understood as presenting {$\mathbb{H}$} as purely even. This can be understood as saying that {$M_2()$} is a purely ephemeral contribution.
See: Todd Trimble. The Super Brauer Group and Super Division Algebras.
Pairing and inverting pairs
Pairing yields odd and even possiblities.
Reversing the ordering of the pairs yields a fourfold periodicity of their parity.
The "teaching" pair plays the role of the pre-frame which becomes the post-frame and vice versa.
Lie group embeddings
Lie algebra embeddings
The Lie algebra embeddings describe divisions of everything as situated between a pre-frame {$\frak{g}_0$} and a post-frame {$\frak{g}$}{$_k$}. Namely, {$\frak{g}_0-\frak{g}_k=\frak{m}_0+\frak{m}_1+\dots\frak{m}$}{$_{\mathit{k-1}}$}.
And every frame {$\frak{g}$}{$_k$} can be further considered as the composition of a perspective {$\frak{m}$}{$_k$} and a frame {$\frak{g}$}{$_{k+1}$}. The linear complex structure {$J_k$} enacts this decomposition as it anticommutes with {$\frak{m}$}{$_k$} (as with rotations times reflections, what we don't know but reflect) and commutes with {$\frak{g}$}{$_{k+1}$} (as with rotations, our direct experience). Thus the frames express our unconscious and they bookend the perspectives that we access with our conscious. Including more perspectives delves deeper into our unconscious to separate direct and indirect experience.
A perspective can be identified with
- The subspace {$\frak{m}_\mathit{k}$}
- The linear complex structure {$J_k$} which commutes with the Lie subalgebra {$\frak{h}_\mathit{k}$} but anticommutes with all of the subspaces {$\frak{m}_\mathit{j}$}, {$0\leq j\leq k$}.
Consider how the Lie group embeddings get expressed by the Lie algebras and their root systems.
Quantum symmetries
Consider the role of antiautomorphisms in quantum symmetries.
Charge conjugation (transposing linear)(mapping particles to holes and holes to particles) expresses what is inaccessible, either unconsciously, implicitly (as with the nullsome) or consciously, explicitly (as with the foursome).
Time reversal (usual antilinear)(mapping particles to particles and holes to holes) expresses what is accessible, either consciously (as with the twosome) or unconsciously (as with the sixsome).
Odd divisions (which leave an unpaired perspective) manifest all three symmetries. Even division (pairing up the perspectives) manifest only one symmetry.