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博特周期性定理

Overviews

• Wikipedia: Bott periodicity theorem
• Raoul Bott. The Periodicity Theorem For The Classical Groups And Some Of Its Applications In 1970, Bott looked back at his original proof, reconsidered it in terms of functors, vector bundles and K-theory, and considered applications, such as parallelizability.
• Math Overflow. Overview of proofs of Bott-periodicity Emphasizes the role of Thom isomorphisms. Asking for the gist of why this should be periodic.

Videos

• Ravi Vakil slides Symmetric spaces and loop space iteration. Complex and real. Focuses on double loop space iteration, using short exact sequences to show that mapping spheres into one space is the same as mapping points into another space.
  • Ravi Vakil "Bott periodicity, algebro-geometrically", part 1
  • Ravi Vakil "Bott periodicity, algebro-geometrically", part 2

Homotopy groups

• John Baez. Bott periodicity. Considering how {$\pi_0(O(\infty))$} relates to {$\mathbb{R}$}, {$\pi_1(O(\infty))$} relates to {$\mathbb{C}$}, {$\pi_3(O(\infty))$} relates to {$\mathbb{H}$} and {$\pi_7(O(\infty))=\pi_{-1}(O(\infty))$} relates to {$\mathbb{O}$} and the Grassmannian.
• Caleb Ji. Various Statements of Bott Periodicity. A short overview of statements in homotopy theory and K theory.
• Jonathan Block. The Bott Periodicity Theorem. First, periodicity allows one to deloop classifying spaces and thus define cohomology theories. Second, using periodicity, “wrong way” functoriality maps can be defined and these are of integral importance in the index theorem. Siebenmann periodicity states that for M a closed manifold {$S(M)∼=S(M×I4,M×∂I4)$}. A map {$S(M)→KO[12]∗(M)$} which intertwines the two periodicities. It should be noted that {$KO[1/2]$} is four periodic with the signature operator playing the same role as the inverse of the Bott element as the Dirac operator does for complex (or real) periodicity.

Loop spaces and suspension

• Tai-Danae Bradley. "One-Line" Proof: Fundamental Group of the Circle.
• Tai-Danae Bradley, Bryson, Terilla. Chapter 6. Paths, Loops, Cylinders, Suspensions... Suspension-loop adjunction.
• Algebraic Topology July 2016 University of Chicago
• Peter May. Introduction to Algebraic Topology 10:30 Freudenthal Suspension theorem. {$X\wedge Y=X\times Y/X\cup Y$}, {$\Sigma X = X\wedge S^1$}, {$S^n\wedge S^1=S^{n+1}$}
• Peter May. An excellent overview of Bott periodicity. Text
  • He goes through Bott's approach. Given symmetric space {$M$} and a triple {$ν = (P, Q; h)$} where {$h$} is the homotopy class of curves joining point {$P$} to {$Q$} in {$M$}. Define {$M^{\nu}$} as the set of all geodesics of minimal length which join {$P$} to {$Q$} and are in the homotopy class {$h$}. Bott shows that {$M^{\nu}$} is a symmetric space. {$\pi_k(M^{\nu})=\pi_{k+1}(M)$} for {$0<k<|\nu |-1$}. Here {$|\nu |$} is the first positive integer which occurs as the index of some geodesic from {$P$} to {$Q$} in the class {$h$}. The index is given by the Morse index theorem.
  • He also considers Hopf algebras as relevant for complex Bott periodicity.
• Stable algebraic topology, 1945-1966 History of algebraic topology in the 1950s. Includes a history of the proofs of Bott periodicity.
• Raoul Bott. What Morse missed by not talking to Weyl Includes Morse theory and loop spaces.
• M.A. Aguilar, Carlos Prieto. Quasifibrations and Bott periodicity. May says this is a modern, simplest proof in terms of homotopy theory. It deals with the complex case.
• Mark J. Behrens. A new proof of the Bott periodicity theorem. A simpler proof of the loop space result, both for complex and real, simplifying and extending ideas of Aguilar and Prieto, based on an appropriate restriction of the exponential map to construct an explicit quasifibration with base space U and contractible total space.

Homology, cohomology

• A question about the topological proofs of Bott periodicity Answer by Peter May. Emphasizes the duality of Hopf algebras.

Symmetric spaces

• John Milnor. Morse Theory. (AM-51), Volume 51 Contains a helpful proof of the Bott periodicity theorem, both real and complex, in terms of Morse theory, minimal geodesics and symmetric spaces. Relates the Lie groups to the symmetric spaces and loop spaces.

Lie group embeddings

• Dyer, Lashof. A topological proof of the Bott periodicity theorems. Loop spaces.

Symplectic structures

• Max Karoubi. Quadratic forms and Bott periodicity. Presents in terms of quadratic forms and symplectic structures.

Morse theory and geodesics

• Loring Tu. The life and works of Raoul Bott.
  • Morse theory
  • Lie groups and homogeneous spaces
  • Index of a closed geodesic Bott considered the critical points of the energy function, which are precisely the geodesics from {$p$} to {$q$}.
  • Homogeneous vector bundles
  • The periodicity theorem
  • Clifford algebras
• Dubrovin, Fomenko, Novikov. Modern Geometry - Methods and Applications. Part III. Introduction to Homology Theory. Includes Bott Periodicity, real and complex, pages 270 to 325. Based on Morse theory, calculus of variations. Also considers the seven dimensional sphere.
• Bosman Honor's Thesis: Bott Periodicity with sketch of proof in terms of Morse theory
• Minimal geodesics (from pode to antipode) parametrize the next space (in the next dimension). In low dimensions, minimal geodesics adequately model loop spaces.
• Langlands program related to Bott-Atiyah. The Yang Mills Equations Over Riemann Surfaces, Morse theory - calculus of variations, thus to Bott periodicity. And Langlands is related to conjugacy classes on GL(2), which are eigenvalues. There are three families of double coverings.

Real, Complex, Quaternions, Octonions

• Eschenburg. Geometry of Octonions Relates the representations of Clifford algebras to the normed division algebras. Identifies the octonions with the representations for {$Cl_{0,4}$}, {$Cl_{0,4}$}, {$Cl_{0,4}$}, {$Cl_{0,4}$} and identifies {$M_2(\mathbb{O})$} with the representations for {$Cl_{0,8}$}. Gives a proof of Bott periodicity in terms of symmetric spaces, symmetric subspaces, poles and centrioles.
• Arkadii Slinko. 1, 2, 4, 8,... What comes next? Parallelizability of spheres.

Octonionic line bundles

• John Baez. OP1 and Bott Periodicity. Canonical line bundles {$L_\mathbb{R}$}, {$L_\mathbb{C}$}, {$L_\mathbb{H}$}, {$L_\mathbb{O}$} give elements {$ [L_\mathbb{R}]$}, {$ [L_\mathbb{C}]$}, {$ [L_\mathbb{H}]$}, {$ [L_\mathbb{O}]$} that generate, respectively, {$\widetilde{KO}(S^1)\cong\mathbb{Z}_2,\widetilde{KO}(S^2)\cong\mathbb{Z}_2,\widetilde{KO}(S^4)\cong\mathbb{Z},\widetilde{KO}(S^8)\cong\mathbb{Z}$}. The isomorphism {$\widetilde{KO}(S^n)\rightarrow \widetilde{KO}(S^{n+8})$} given by {$x\rightarrow [L_\mathbb{O}]$} yields Bott periodicity. The canonical octonionic line bundle over {$\mathbb{O}P^1$} generates Bott periodicity.

Clifford modules

• Dexter Chua. Clifford Algebras and Bott Periodicity.
  • Bott Periodicity Seminar
  • Deke Zhao. Graded Morita Equivalence of Clifford Superalgebras explains how to calculate the graded representations and the Grothendieck groups.
    • Math Stack Exchange. Tensor product of graded algebras.
  • M.F.Atiyah, R.Bott, A.Shapiro | Clifford Modules.
  • H. Blaine Lawson, Marie-Louise Michelsohn. Spin geometry.
• Yuqin Kewang. Clifford Algebras and Bott Periodicity.
• Andre Henriques. A proof of Bott periodicity via Clifford algebras
• Baez: 2) H. Blaine Lawson, Jr. and Marie-Louise Michelson, "Spin Geometry", Princeton U. Press, Princeton, 1989. or 3) Dale Husemoller, "Fibre Bundles", Springer-Verlag, Berlin, 1994. These books also describe some of the amazing consequences of this periodicity phenomenon. The topology of n-dimensional manifolds is very similar to the topology of (n+8)-dimensional manifolds in some subtle but important ways!'' Physics of fermions.
• Dale Husemoller's book is detailed and helpful.
• Roberto Rubio. Clifford algebras, spinors, and applications. Classification of Clifford algebras.

Representations

• Bosons - real representations, fermions - quaternionic representations.
• Spin representation

Spinors

• Cameron Krulewski. K-Theory, Bott periodicity, and elliptic operators Explains Attiyah's proof which uses the index of elliptic operators. The real case is the spinor case and mentions Dirac operators.

Morita equivalences as quantum Hamiltonian reductions

• Theo Johnson-Freyd. The Quaternions and Bott Periodicity Are Quantum Hamiltonian Reductions.
• We show that the Morita equivalences Cliff(4) ' H, Cliff(7) ' Cliff(−1), and Cliff(8) ' R arise from quantizing the Hamiltonian reductions R0|4//Spin(3), R0|7//G2, and R0|8//Spin(7), respectively.
• The Morita equivalence {$\textrm{Cliff}(7) \simeq \textrm{Cliff}(−1)$} arises from the Hamiltonian reduction {$\mathbb{R}^{0|7}//G_2$}, where {$G_2 ⊆ SO(7)$} is the exceptional Lie group of automorphisms of the octonion algebra {$\mathbb{O}$}.
• Hamiltonian Reduction by Stages

CPT symmetry

• Andrzej Trautman. On Complex Structures in Physics. Baez discusses this. Trautman relates Clifford algebra clock, spinors and charge conjugation.

Unit spheres

• John Baez. Octonions, normed division algebras and Bott periodicity. Clifford algebras {$Cl_{0,n}$}, whether they admit a representation on {$k$}-dimensional vector space, which is if and only if the unit sphere in that vector space {$S^{k-1}$}, admits {$n$} linearly independent smooth vector fields.
• Max Karoubi. Algebraic maps between spheres and Bott periodicity.
• Related to Adams's work on vector fields on spheres.
• 6+4=2 modulo 8

24-cell?

• John Baez's talk on the 24-cell, video, slides

Extensions

• Jim Bryan, Marc Sanders. Instantons on {$S^4$} and {$\cpbar$}, rank stabilization, and Bott periodicity
• Math StackExchange: 8 periodicity: Clifford clock- Bott periodicity - KO-dimension in noncommutative geometries
• Daisuke Kishimoto. Topological proof of Bott periodicity and characterization of BR. (1,) periodicity

Ideas

• Max Karoubi mentioned in his video that loop equations for rings with R,C,H,H' and {$\epsilon = +/-1$} yields 10 homotopy equivalences.
• Orthogonal add a perspective (Father), symplectic subtract a perspective (Son).
• John Baez's comment about 4 dimensions being most interesting because half-way between 0 and 8. The Son turns around and reverses the Father.
• In Bott periodicity, the going beyond oneself (by adding perspectives) and the going outside of oneself (by subtracting perspectives) is, in mathematics, extended to all possible integers, and thus the two directions are related in the four ways as given by the four classical Lie algebras, including gluing, fusing, folding.
• In the eight-cycle of divisions, the going beyond oneself and the going back outside oneself are reminiscent of spinors like electrons.
• The flip side of going beyond yourself - if you can add perspectives, then you can substract perspectives - on the flip side. Thus nullsome and foursome model the extremes: nullsome models the void of everything, foursome models the point of nothing. Thus adding a perspective takes us from the indefinite to the definite and back again. The three operations +1,+2,+3 make for a three-fold braid.
• Goedel's incompleteness theorem. There is irrelevant, inaccessible knowledge, such as that which describe different implementation of equivalent rings (for Bott periodicity) where the equivalence means that they have the same (isomorphically, structurally) representations.
• Bott periodicity exhibits self-folding. Note the duality of 1 with the pseudoscalar.

Raoul Bott

• Raoul Bott mirė Carslbad, Kalifornijoje. Attiyah apie Bott gyvenimą. Nekrologas New York Times Dukra Jocelyn Scott gyveno Rancho Santa Fe, dirbo C.B.S. Scientific. Jisai gyveno: assisted living - Sunrise at La Costa, 7020 Manzanita St, Carlsbad, CA 92011-5123 Studied geodesics on {$SU(2)$}.

People who may care

• Tyler Marshall Goldstein. Stages of sentience (Bott periodicity)
  • Tyler Goldstein at Linked In I contacted him.
• The Larson Research Center. Bott Periodicity Theorem.

Renormalization group: Block spin This model is relevant when everything has only one parent, when the matrices are sparse and so there are local collections as to causality. Similarly, the chance of two neurons being connected is very small. Relate this to Bott periodicity. Each level increases or decreases the level of detail. But eight levels of detail (eight contexts) are all that you can distinguish and then you would just repeat.

In Todd Trimble's classification of super division algebras, consider how they relate to the Chomsky hierarchy. In particular, for {$H+He$} we have {$hex=exh$} as with a linear Turing machine by which we could have {$hhhhhhex=exhhhhhhh$}. For {$C+Ce$} we have {$ce=e\bar{c}$}. This means that (when {$|c|=1$}) we have {$e=cec$} as with a pushdown automata. For {$R+Re$} we have {$re = er$}. If {$r=1$}, then {$e=er$}, as with finite automata. So it's true modulo real scalars. And then we should have a collapse {$e=$} disappears, as with {$C\ell_{\pm 4}$} when {$H$} is just the even part.

Think of Bott periodicity as 8-2=6 where the -2 subtracts the complex Bott periodicity, which represents God and good in 6+1+1 and which creates the context for contradiction, making for the Bott cycle.

Odd is reflective, switching spaces to acknowledge the reflection. Even is nonreflective (raw).

How does time arise from the symmetry of time reversal? Does unconscious switch the direction? (going from effect to cause?) and conscious switch it a second time, thus keeping it the same? (from cause to effect?) Knowing becomes the past (decreasing slack). Not knowing opens up the future (increasing slack).

The Clifford algebra expression {$\cos\theta e_1+\sin\theta e_2$} for the pin group {$Pin_\pm(2)$} may express the perception-action loop. It is periodic like a photon.

• Time reversal determines the context, whether the division algebras are real ({$T^2=+1$}), complex (no {$T$}) or quaternion ({$T^2=-1$}). This distinguishes how many perspectives there are. Whereas the perspective itself is given by charge conjugation, with either {$C^2=-1$} (conscious) or {$C^2=+1$} (unconscious).
• Over the reals the Clifford algebras {$Cl_{n,0}$} and likewise {$Cl_{0,n}$} are matrix superalgebras if n is divisible by 8. Their elements are either even (diagonal blocks) or odd (off diagonal blocks). Calculate this explicitly.

The linear complex structure {$ \begin{pmatrix}0 & -1\\1 & 0 \\ \end{pmatrix}$} squares to {$-1$} and has order {$4$} and models the conscious outlook. Whereas its counterpart in the representation of {$CT$}-groups is odd antilinear {$ \begin{pmatrix}0 & 1\\1 & 0 \\ \end{pmatrix}$}, squares to {$1$}, and models the unconscious outlook.

• Compare the understanding of Bott periodicity in terms of quantum symmetries with the understanding in terms of mutually anticommuting linear complex structures.
  • The first linear complex structure is {$\bar{C}$} and then we add, in turn, changing the context, {$i$} (to get {$i\bar{C}$}) and {$\bar{T}$} (to get {$\bar{C}\bar{T}$}.

The length of quaternions models the independence of two planes. That is possible in three different ways.

How is {$\mathbb{H}\oplus\mathbb{H}$} related to the eightsome?

How is {$\mathbb{H}(2)$} related to the eightsome?

In Bott periodicity, what does it mean that adding generators which square to -1 and adding generators which square to +1 yields the same when we have four generators? Is that (unconscious) knowing and (conscious) not knowing? Are they complementing each other with regard to the eightsome?

Is there (unconscious) contextualizing and (conscious) decontextualizing (deconstructing)? And does this yield contexts as the manifestation of consciousness? What has priority?

A gapped Hamiltonian can be understood as {$H\neq 0$} separating and distinguishing {$H>0$} (conscious) and {$H<0$} (unconscious). This establishes a change in perspective, stepping in and stepping out.

The three-cycle is a parity operator (for space) which establishes supersymmetry. Space and time together define the foursome.

Knowledge is the marking of a state in a symmetry. You have two views (two levels). Different states (known) and their symmetric transformation (no visible change, thus unknown).

The essence of Bott perioidicity is relating (in terms of their actions, their symmetries) a (one-dimensional) circle {$U(1)$} (embedded in two dimensions) and {$\mathbb{Z}_2\times\mathbb{Z}_2$} (two zero-dimensional circles embedded in two dimensions)(or a vertical strip with top and bottom, front and back)

Channels in reservations

• 0 channels {$\mathbb{R}\oplus 0$} nullsome (the zero)
• 1 channel {$\mathbb{R}\oplus\mathbb{R}$} onesome
• 2 channels {$\mathbb{C}\oplus\mathbb{C}$} twosome
• 3 channels {$\mathbb{H}\oplus\mathbb{H}$} threesome
• {$(\mathbb{H}\oplus\mathbb{H})\oplus (\mathbb{H}\oplus\mathbb{H})$} foursome {$\mathbb{H}\oplus 0$}

The circle {$U(1)$}, key for quantum symmetry, models a perspective, is the source of rotations, the foundation of geometry.

Linear (this is this; that is that) and antilinear (this is that; that is this).

Linear and antilinear - Bott periodicity - how the question and answer relate. Does the answer follow the question? Or does the question follow the answer?

• complex Bott periodicity models the being and real Bott periodicity models their mind (and their three minds)
• division algebras R,C,H are the even parts of super division algebras where {$e=0$} thus {$e^2=0$}.
• what is real is unreflected
• {$-I$} as reflection. {$J_1J_2=-J_2J_1$} One way is the reflection of the other.
• How is {$BSp$} related to the three minds, +3? and +2?
• What is {$BO$}? How is it related to {$O$}? and what does {$\mathbb{Z}$} mean here?
• Physics distinguishes observer and observed like even and odd parts of a superalgebra (or like context and experience?) And the automorphism is slack, which can be increasing or decreasing.
• the foursome relates {$V_+$} and {$V_-$} as two twosomes. And then divide further into two. A perspective becomes a twosome. Until it all collapses.

Orthogonality appears in Bott periodicity. Linear complex structures are orthogonal skew-symmetric matrices (for which {$j^T=-j$} so that together {$j^{-1}=-j=j^T$}. My approach to the wave function is based on orthogonal Sheffer polynomials. What is the relation, if any, between Sheffer polynomials and skew-symmetric matrices?

The symmetry of Bott periodicity does not depend on the size of the dimensions but is deeper than whole numbers. If we think in terms of numbers, then it becomes much more complicated, as with the homotopy of spheres.

In Bott periodicity, generators squaring to +1 give quaternionic structure? How does that equal to the operator +3 for consciousness with linear complex structures?

Noncommuting means incompatible frameworks. So rotations and rotoreflections are incompatible. Unconscious and conscious are incompatible. But there is an isometry between unconscious and conscious.

The fact that, for orthogonal matrices, {$O^T=O^{-1}$} and thus the inverse can be understood as reversing arrows (similarly for the unitary and compact symplectic cases) is an expression of the symmetry in math. Bott periodicity organizes these symmetries in math itself.

Is there a category where the arrows are products of linear complex structures and the objects are subgroups of the orthogonal group? Does composition work out?

The (nonassociative) octonions are perhaps related to the (associative) split-biquaternions, as the Lie group embedding unfolds, by considering that we don't know whether we Humans are living in the reflected world (based in God's mind) or unreflected world (based in our self), in the chain Human's view of God's view of Human's view... We may not know if we are living in the {$V_+$} or {$V_-$} of the split-biquaternions. This may give rise to an ambiguity, and also, a choice between our God and our self. That choice and that ambiguity may ground nonassociativity of perspectives, distinguishing between stepping out [Child's view of (Father's view of Mother's view)] and stepping in [(Child's view of Father's view) of Mother's view]. When associativity no longer holds, the system collapses, and we have zero objectively, and perhaps the field with one element, subjectively.

The even subalgebra of a Clifford algebra {$Cl_{0,n+1}$} is the Clifford algebra {$Cl_{0,n}$}. The unit elements of that Clifford algebra is the spin group {$Sp(n+1)$}.

Think of divisions of everything as an adjoint functor to the application of a linear complex structure on a symmetric space.

In what sense is a linear complex structure the going beyond of oneself?

{$J_1$} is a linear complex structure which models the unconscious, adding a perspective {$+1$}. {$J_2$} adds an antilinear operator which models reflection and a perspective on a perspective. Together they yield the conscious. {$J_3$} divides the space on which these act into two parallel spaces, thus is a perspective upon a perspective on a perspective. Together they are consciousness. {$J_3J_4$} defines an isometry between the two spaces.

• +1 Add one operator. Impose linear complex structure. I. Unconscious.
• +2 Add two mutually anticommuting operators. Impose quaternionic structure. You. Conscious.
• +3 Add three mutually anticommuting operators. Impose split-biquaternionic structure. Other. Consciousness.

Linear complex structure {$J_i$} is a perspective. A shift in perspective from {$J_i$} to {$J_j$} is their product {$J_iJ_j$}. Together they define a quaternionic structure. What is the meaning of longer products of perspectives?

Freeman J. Dyson. The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics. J. Math. Phys. 3, 1199–1215 (1962)

Symmetry breaking distinguishes vectors and spinors.

We can carve up mental space from either end, in the (+1) and (-1) directions, or both ends (+2), arriving at {$Cl_{4,0}\cong Cl_{0,4}$}. How to understand (+3)? Compare Clifford algebras with chain complexes.

{$A_1$} is the building block for Lie theory and its Lie algebra serves both {$SU(2)$} and {$SO(3)$}, thus relates vectors and spinors, and seems relevant for Bott periodicity. Could the exceptional Lie groups {$E_k$} be generated by Bott periodicity?

A symmetric space has, at each point, a (global) isometry that (locally) inverts each tangent vector. Compare that with symmetry where we reflect across the center of a space.

In John Baez's talk on the symmetric space, the functor acts on the category of representations of Clifford algebras, which is the category relevant for Morita equivalence.

Doubling, halving, dividing a perspective (a space, a module) into two perspectives, yielding a perspective on a perspective.

In a complex Clifford algebra, the coefficient {$i$} commutes with all of the generators {$e_k$} whereas the generators anti-commute with each other. If the coefficient {$i$} anticommuted with all of the generators, then this would simply be a real Clifford algebra with one more generator. So this is a very deep fact that distinguishes complex and real Clifford algebras.

Think of Clifford algebra generators that square to {$-1$} as spinors, and those that square to {$+1$} as vectors. Each generator can be thought of as yielding a turn of {$\pi=180^\circ$}. Does that make sense?

Rulead space generated by three minds, the three possible rules on the eight cycle.

Are the CPT symmetries related by the three-cycle? What combinations are possible?

• https://math.stackexchange.com/questions/2230802/how-to-prove-that-a-ring-r-is-morita-equivalent-to-m-nr
• Consider the ways of constructing a 2x2 matrix by adding a generator {$a_i$} and then {$b_j$}, or alternatively, {$b_j$} and then {$a_i$}.

Is the {$E_8$} symmetry based on one half of Bott periodicity? Do the Dynkin diagrams of the exceptional Lie groups model counting backwards and forwards inside Bott periodicity?

Neil Turok Mirror universe, big bang as a mirror, CPT symmetry.

Is simultaneous CPT transformation the same as switching us from rotations to reflected rotations? Are there two parts of {$O(\infty)$}? And is this like the unconscious and the conscious, where the conscious is a reflected version of the unconscious?

Barratt-Priddy-Quillen theorem. The group completion of the monoid of finite sets {$Fin^{gp}$} is the stable homotopy group of spheres.

Classifying space for U(n) is the complex Grassmannian. Classifying space for O(n) is the real Grassmannian.

{$O(\infty)$} has two parts and that may be reflected in the fact that the simple roots can be considered in two groups {$x_i-x_j$} and {$x_j-x_i$} where {$i<j$}. Consider how this works for each Lie group, unitary, symplectic, odd and even orthogonal groups.

Grassmannian G(4,2) has five Plucker coordinates (4 choose 2) minus 1. Because it is up to scaling, which gives an extra dimension. Four points on a circle - if two chords cross, they can be untangled in two ways. Cluster algebra and "mutations" by which one product is the sum of two products. Ptolemy theorem.

Idea: The threesome is what links together the two worlds of O(infinty). The threesome equates a shift in one world with a node in another world and vice versa. And this creates a circle - the three-cycle - which moves in one direction - distinguishing what is unconscious and what is conscious and defining a hole for Z.

Basis is a context, supplies a context, provides a context. The number of bases involved is the context.

Reflection is not commutative with regard to a linear complex structure.

In the orthogonal group, commuting means that we have some eigenvector.

• What does it mean if operators are anti-commuting?

Consider all permutations of all lengths. (In considering the moments.) This is similar to {$O(\infty)$} or {$U(\infty)$} which considers all rotations. There is a theorem that relates them. And consider how to take the Fourier transform of all of that. And how does that relate to the Fourier transform of finite groups?

How does matter antimatter asymmetry relate to the two branches of {$O(\infty)$} and the weak force?

Spheres make prominent the dimensions 0/1, 1/2, 3/4, 7/8 that accord with the normed division algebras. The divisions of everything likewise pair odd and even divisions but also include 5/6. So is there any connection or not?

nLab: Sphere spectrum The sphere spectrum is the suspension spectrum of the point. The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres. The sphere spectrum is the higher version of the ring Z of integers. See also: nLab: Suspension

Assignment is asymmetric (computer science), equality is symmetric (math). Compare this with linear complex structure.

{$S_\infty$} = colimit of symmetric groups = permutations with finite support

The conjugate of a quaternion flips the sign of all three dimensions. Is this a manifestation of a parity transformation?

Could twistors relate the two branches of {$O(\infty)$}? And how could the (left-handed or right-handed) chirality of spinors relate to those two branches?

How do the transpose, conjugate transpose, quaternionic transpose simplify Cramer's rule combinatorially?

• We keep {$T$} fixed and change {$C$} proceeding {$-1,0,1$}. This brings to mind the sevensome, nullsome, onesome. Here the nullsome means the nonexistence of the symmetry. Similarly, we keep {$C$} fixed and change {$T$}.