Spin representations, Spin representation periodicity
Andrius: I'm investigating:
How to model divisions of everything with spin geometry?
I want to interpret the Clifford action and then the isomorphism between this action (this representation) and its dual action (its dual representation).
The {$n$}-dimensional vector space {$V$} is generated by {$n$} perspectives. They are paired up to give {$m$} pairs {$(a_j,\bar{a_j})$} with possibly one perspective {$u$} left over. The vectors {$a_j$} generate {$W$}, {$\bar{a_j}$} generate {$W'$}. They can be identified with the mind that does not know, which defines contexts (for what is not), and the mind that knows, which defines concepts (for what is).
The spin algebra (the bivectors {$\wedge^2 V$}) act on the spin structure {$S=\wedge^\bullet W$} and likewise on the dual spin structure {$S^*=\wedge^\bullet W'$}.
If {$V$} is zero-dimensional or one-dimensional, then the bivectors {$\wedge^2 V=0$}.
If {$V$} is even-dimensional, then the action of its Clifford algebra equals linear actions upon the exterior algebra of {$W$}, of what is.
If {$V$} is odd-dimensional, then it includes a reflective third mind which reflects the sign, and then the action of the Clifford algebra equals two separate actions, one on {$S$}, in terms of what is, the other on {$S'$}, in terms of what is not, with the action of {$u$} relating the two.
So the perspectives of a division of everything are understood in terms of their actions on the shifts in perspective within that division of everything.
The spin structure generates all subsets of shifts in perspective. {$W$} identifies the {$j$}th shift in perspective with its intial perspective {$a_j$}, and {$W'$} identifies it with its final perspective {$\bar{a_j}$}.
The action is by bivectors of the form
- {$a_j\wedge \bar{a_j}$} If the set includes the shift, then keep the set as is. (Double check regarding sign, etc.) If the set does not include the shift, then output {$0$}.
- {$\bar{a_j}\wedge a_j$} If the set includes the shift, then output 0. If the shift does not include the shift, then include it, along with sign for positioning.
- {$a_j\wedge \bar{a_k}$} If the set does not include {$k$}, or if it includes {$j$}, then output {$0$}. Otherwise, replace shift {$k$} with shift {$j$}, adjusting signs as needed.
- {$\bar{a_j}\wedge a_k$} If the set includes {$k$}, or if it does not include {$j$}, then output {$0$}. Otherwise, replace shift {$j$} with {$k$}, adjusting signs as needed.
- {$a_j\wedge a_k$} If the set includes {$k$} or {$j$}, then output {$0$}. Otherwise, add {$k$} and then {$j$}, with signs as needed.
- {$\bar{a_j}\wedge \bar{a_k}$} If the set does not include {$j$} or {$k$}, then output {$0$}. Otherwise, remove both {$k$} and {$j$}, adjusting signs as needed.
- {$a_j\wedge u$} If the set includes {$j$}, then output {$0$}. Otherwise, replace all generators with their negatives, and then add {$j$}, adjusting signs as needed.
- {$u\wedge a_j$} If the set includes {$j$}, then output {$0$}. Otherwise, add {$j$}, adjusting sign as needed, and then replace all generators with their negatives.
- {$\bar{a_j}\wedge u$} If the set does not include {$j$}, then output {$0$}. Otherwise, replace all generators with their negatives, and then remove {$j$}, adjusting signs as needed.
- {$u\wedge\bar{a_j}$} If the set does not include {$j$}, then output {$0$}. Otherwise, remove {$j$}, adjusting sign as needed, and then replace all generators with their negatives.
In general, we are adding or removing two shifts in perspective, or replacing one with another. Or instead of a shift we are reflecting the signs.
This suggests that the action of the bivector Second {$\wedge$} First relates the three minds. The third mind is given by {$u$}. Alternatively, it describes shifts in shifts in perspective.
We want to understand the isomorphism between the representation {$S$} and the dual representation {$S^*$}.
Notes
The differentiation of an curve {$\gamma (t)$} can be understood as an operation on the manifold. Consider a circle. Then if the curve is expressed as going around a circle, then differentiation makes for a rotation by {$\frac{\pi}{2}$}. Alternatively, consider a line. If the curve goes along the line, then differentation makes for a istropic matrix of the form {$a_j$}. Note also that we can interpret the two ways of presenting the special orthogonal group as expressing its curve as a rotation around a circle or as a translation along a line (perhaps in the complex plane).
The {$a_j$} and {$\bar{a_j}$} are interpreted as fermion creation and annihilation operators. (Sattinger, Weaver)
Concepts
- The chain of perspectives {$PPP$} is modeled by reversion and the chain of Human's and God's perspectives {$HGHGH$} is modeled by conjugation.
Odd {$n=2m+1$}
{$V=W\oplus W'\oplus U$} has an extra dimension given by {$U=\langle u \rangle$}.
Even {$n=2m$}
There are two irreducible spin representations and they model the two minds.
{$n=8k$}
In the case of {$SO(8)$} there is triality which I expect models the three minds.
Pre-frame and post-frame
A division of everything supposes that it is experienced from the vantage point of a pre-frame and also a post-frame. These relate to the two minds. A division of everything is experienced through the subset of its shift-in-perspectives for which the shift occurs, which means there is a complementary subset of shifts which do not occur. The pre-frame is the vantage point before these shifts-in-perspective occur. The post-frame is the vantage point after these selected shifts-in-perspectives occur.
The pre-frame is given by the Lie algebra {$\frak{g}{$_0$} and the post-frame for the division of everything into {$k$} perspectives is given by the Lie algebra {$\frak{g}$}{$_k$}. We have {$\frak{g}{$_0-$}{$\frak{g}{$_k =+\frak{m}{$_0 +\cdots + \frak{m}{$_0$}.
We can think of the pre-frame and post-frame as an additional pair of dancers, the teaching pair. This should make the numbers work out correctly.
The pre-frame and post-frame can perhaps be identified with the two conceptions of divisions (for the foursome, fivesome, sixsome, sevensome) where there are two or three shifts in perspective. Whereas when there are but zero or one shifts in perspective, we should have four conceptions.
Note that the bilinear form, when skew, has us switch minds (answering or questioning) when we go from pre-frame to post-frame. Whereas a symmetric form has us stay within the same mind.
Furthermore, we may split the shifts among two minds, when the representation is reducible. This yields two channels.