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Andrius Kulikauskas

  • m a t h 4 w i s d o m - g m a i l
  • +370 607 27 665
  • My work is in the Public Domain for all to share freely.

用中文

  • 读物 书 影片 维基百科

Introduction E9F5FC

Questions FFFFC0

Software

Bott Periodicity Models Divisions of Everything


Model various features of divisions

  • Perspective
  • Shift in perspective
  • Three minds +1, +2, +3
  • Three-cycle
  • The two axes of the foursome, fivesome, sixsome, sevensome and the gap between the axes
  • Collapse of the eightsome into the nullsome
  • 24 equations

Make sense of various aspects of Bott periodicity

  • The homotopy groups {$\mathbb{Z}_2$}, {$\mathbb{Z}$}, {$0$}
  • C, P, T symmetries
  • Linear complex structure, quaternionic structure, split-biquaternionic structure
  • Twofold Bott periodicity
  • The splitting of the Lie groups {$Sp(4), Sp(2)\times Sp(2), Sp(2)$}; {$O(2), O(1)\times O(1), O(1)$}; {$U(2),U(1)\times U(1), U(1)$}.

Divisions of everything

  • Morse theory - count the number of directions of steepest ascent - does this relate to the divisions of everything?
  • Eight-cycle of divisions explains how noncontradiction arises from contradiction, thus it is important for life and for consciousness, for preservation and transformation of the whole. How does this relate to decreasing entropy?
  • Can we think of {$S^n$} as relating {$n$} dimensions in a division of everything? With dimension growing per Bott periodicity, as with topological insulators?
  • In Bott periodicity, do the real numbers express God (the nullsome)? and the quaternions express knowledge (the foursome)?
  • In what sense is Bott periodicity the paradigm for recurring activity in Christopher Alexander's patterns? Is it related to God going beyond himself in four steps and human going beyond himself in four steps back to God, on the complementary side of the divisions of everything?

Lie group inverses

  • Interpret the entries of the inverse of a compact symplectic matrix.

Six transformations

  • Compare the related Lie groups (and their connections with spheres) to the six specifications of geometry, the six transformations of perspectives

Adjunctions and Conceptions

  • Consider the role of the loopspace - suspension adjunction.
  • Look for links to Grothendieck's six functor formalism and my classification of adjunctions.

I am summarizing my investigation of how Bott periodicity models the divisions of everything.

The most promising approach is to consider the embeddings of Lie groups.

A perspective is modeled by a linear complex structure {$J_k$}. These act on an orthogonal group {$O(16r)$} where {$r$} is an integer. {$J_k^2=-1$}.

Distinct perspectives {$J_j$} and {$J_k$} are mutually anticommuting linear complex structures, which is to say, {$J_jJ_k=-J_kJ_j$}.

Thus a set of {$k$} distinct perspectives is understood as the generators of the Clifford algebra {$Cl_{0,k}$}.

I am trying to identify a set of {$k$} linear complex structures {$J_1, J_2,\dots ,J_k$} with the division of everything into {$k$} perspectives.

As we apply these structures sequentially to {$O(16r)$}, we can ask, which matrices they commute with, and which do they not commute with? In general, it seems that they commute with those which have no reflection, and they do not commute with those which have a reflection. But this notion changes as we keep applying a new, mutually anticommuting structure.

Mental reflection is modeled variously, as inversion, as reflection, as complex conjugation, as antilinearity, as that which is culled away because of noncommutativity.


Three Minds

{$J_\alpha J_\beta$} relates Unconscious and Conscious.

{$J_\alpha J_\gamma J_\delta$} divides the vector space into worlds where {$J_\alpha$} and {$J_\beta$} are experienced as How {$-v\rightarrow v$} and What {$v\rightarrow v$} (as in the How and What of an arrow by the Yoneda embedding). Whereas {$J_\gamma J_\delta$} is their framework which is not experienced but grounds the experience as Why and Whether.

The homotopy groups {$\mathbb{Z}_2$} link the second mind with the first mind, and the third mind with the second mind.

The fourth mind is linked to itself, yielding {$\mathbb{Z}$}. So is the eighth mind.


Consider Clifford algebras {$Cl_{0,k}$} generated by {$J_1,J_2,\dots ,J_k$}.


{$Cl_{0,0}=\mathbb{R}$} Nullsome. {$O(16)$} to {$O(16)$}


{$Cl_{0,1}=\mathbb{C}$} Onesome. {$U(8)$} to {$O(16)$}

Any linear complex structure functions as a perspective.


{$Cl_{0,2}=\mathbb{H}$} Twosome. {$Sp(4)$} to {$O(16)$}

Two mutually anticommuting linear complex structures define a quaternionic structure.

A product of two linear complex structures functions as a shift in perspective.


{$Cl_{0,3}=\mathbb{H}\oplus\mathbb{H}$} Threesome. {$Sp(2)\times Sp(2)$} to {$O(16)$}

Pseudoscalar squares to {$+1$}. {$(J_1J_2J_3)^2=1$}

We have that {$J_1J_2J_3=J_2J_3J_1=J_2J_3J_1$}.

We also have that {$J_1J_2J_3$} splits {$V=V_+\oplus V_-$} where {$J_1J_2J_3v_+ = v_+$} for {$v_+\in V_+$} and {$J_1J_2J_3v_- = -v_-$} for {$v_-\in V_-$}.

{$J_1J_2v_- = J_3v_-,\; J_2J_3v_-=J_1v_-,\; J_3J_1v_-=J_2v_-$}

A product of three linear complex structures breaks up the vector space into two eigenspaces, {$S_+$} and {$S_-$}.

Three mutually anticommuting linear complex structures define a split-biquaternionic structure.


{$Cl_{0,4}=M_2(\mathbb{H})$} Foursome. {$Sp(2)$} to {$O(16)$}

{$J_3J_4$} sets up an isometry between {$V_+$} and {$V_-$}.


{$Cl_{0,5}=M_4(\mathbb{C})$} Fivesome. {$U(2)$} to {$O(16)$}

{$J_1J_4J_5$} commutes with {$J_1J_2J_3$} and acts within {$V_+$} (or {$V_-$}) and divides it into mutually orthogonal eigenspaces {$W_\pm$} with {$W_-=J_2W_+$}. {$J_2$} interchanges the {$1$} and {$j$} so we are left with those matrices that only mix {$1$} and {$i$}, thus are complex.

Can we think of the new operators as developing the gap in between? Or can we think of them as appearing on either end of the chain?


{$Cl_{0,6}=M_8(\mathbb{R})$} Sixsome. {$O(2)$} to {$O(16)$}

{$J_2J_4J_6$} commutes with {$J_1J_2J_3$} and with {$J_1J_4J_5$} and acts within {$W_\pm$} and splits it into {$X_\pm$} such that {$X_-=J_1X_+$}. {$J_1$} acts as {$i$}, thus we are left with those matrices that are without {$i$}.


{$Cl_{0,7}=M_8(\mathbb{R})\oplus M_8(\mathbb{R})$} Sevensome. {$O(1)\times O(1)$} to {$O(16)$}

{$J_1J_6J_7$} commutes with {$J_1J_2J_3$}, {$J_1J_4J_5$}, {$J_2J_4J_6$} and splits {$X_+$} into subspaces {$Y_\pm$}


{$Cl_{0,8}=M_{16}(\mathbb{R})$} Eightsome. {$O(1)$} to {$O(16)$}

{$J_7J_8$} commutes with {$J_1J_2J_3$}, {$J_1J_4J_5$}, {$J_2J_4J_6$} but anticommutes with {$J_1J_6J_7$} and thus sets up an isometry between {$Y_+$} and {$Y_-$}. Are there circumstances where this could be considered contradictory?

Consider flavors of Bott periodicity where this collapses back to the nullsome. In particular, consider Morita equivalence. Yet Morita equivalence equates all matrix algebras of a division algebra, thus equates too many structures.


Modeling physical systems

Modeling biological systems

Krebs cycle.

Bott periodicity. Sulfur as an eight-cycle. The origin of life.

Ideas

  • The relevant Lie groups are all rotations about a fixed origin. That fixed origin represents a universal, absolute perspective, God's perspective upon everything, God's knowledge of everything.
  • Divisions of everything are perhaps chopping up a sphere where the sphere is everything also circle folding
  • Bott periodicity should be related to the collapse of the eightsome into the nullsome, and thus the definition of contradiction
  • Perspective arises because of base point - there is a fixed point for the isometries. We are that fixed point.
  • Understand the dimensions of a Lie group as perspectives. And look at Lie groups as rotations of a sphere.
  • The widgets at the end of the Dynkin chain allow for turning around and thus for ambiguity in the understanding of the perspective of the chain and thus they can be understood as divisions of everything. And this should relate to Bott periodicity.
  • ketverybė: H, nulybė: R
  • 2-periodicity: Dievas - gerumas - Dievas
  • sukeisti buvimą ir nebuvimą, tapimą ir netapimą
  • 10 = 4+6 = 4+(4+2) = 8 + 2 (2 tai laisvumo atvaizdai)
  • Z 0 Z 0 Z 0
  • Z 0 (replace Z 0 with Z2 Z2 particle) Z 0 (replace Z 0 with 0 0 hole)
  • 8 external dimensions (adding 4 dimensions or +2 or +3)
  • internally expressed dimensions - adding integrity
  • same pattern reversed internally
  • Equivalence up to isomorphism is relevant for "user requirements" as opposed to "material implementation". This is relevant for Bott periodicity.

Bott periodicity provides order to the homotopy groups of spheres and that reflects that we are left with Consciousness when we unplug the Unconscious and likewise unplug the Conscious. So the relation between n and m is the relation between the Unconscious and the Conscious and their content.

Bott periodicity expresses the symmetry of mathematics

  • compare this with how Lie algebra root systems express the symmetry in counting forwards and backwards, for example, in one direction {$e_i-e_j$} and in the opposite direction {$e_j-e_i$}.

In Bott Periodicity, the matrix representations are related:

  • {$\mathbb{Z}_2$} as a product (limit) expressing external relationships {$\begin{pmatrix}A & \\ & B \end{pmatrix}$}
  • {$\mathbb{Z}$} as a sum (colimit) {$A+B$} and {$A-B$} in terms of internal structure
  • {$0$} representations are the same size. What is the significance?

Loop spaces

  • Think of loopspace as expressing a space for learning.
  • Loop - open arc - perspective - defines inside vs. outside - the ambiguity of a point.

Complex Bott periodicity, which relates U(n) and U(n+m)/U(n)xU(m), and goes back and forth between them, suggests that real Bott periodicity goes back and forth between O(n) and O(n+m)/O(n)xO(m). In one direction there is one step but in the other direction there are seven steps.

In Bott periodicity, {$\mathbb{Z}_2$} may count the unconscious and the conscious. They are the two pieces of {$O(\infty)$}. The homotopy groups explain the different ways of relating them.

A possible interpretation of the sequence. {$\mathbb{Z}$} expresses a hole as with the threesome and the sevensome. {$\mathbb{Z}_2$} describes the nullsome and the onesome.

  • Three minds are three shifts adding 1, 2, 3 dimensions in the Bott clock shift.
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This page was last changed on June 14, 2024, at 03:59 PM