概论

数学

发现

Andrius Kulikauskas

  • ms@ms.lt
  • +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

Software


See: Math, Divisions, NSpheres, Bott Periodicity concepts, The Tenfold Way, String theory, Clifford algebras, Homotopy groups

Investigation: Relate Bott periodicity to the eight-cycle of divisions of everything.


博特周期性定理


  • Compare the related Lie groups (and their connections with spheres) to the six specifications of geometry, the six transformations of perspectives
  • Max Karoubi savo video paskaitoje paminėjo loop lygtį žiedams kurioje R,C,H,H' ir epsilon = +/-1 gaunasi 10 homotopy equivalences. Kodėl 10? 8+2=10? ar 6+4=10, dešimt Dievo įsakymų?
  • How might the fourfold periodicity of the sign of the pseudovector be related to the fourfold periodicity of the differentiation of sine and cosine functions?
  • Does the constraint {$J^2=−I_n$} on complex structures and their anti-commutativity relate to the constraints on Clifford algebras?
  • Relate Bott periodicity with the n-spheres from the 0-sphere to the the 7-sphere (and 8-sphere).

Videos

Statement

Expositions

Extensions

Proofs

Bott-periodicity also arises in string theory.

Related concepts

Math facts

  • Two types of results
    • Periodicity of the homotopy groups of the unitary groups.
    • Periodic theorem of vector bundles.
  • Two-cycle
    • Start from C - as the basic duality - and end up at C inverted (perhaps the conjugate) and go back. This circle is used to list the quotients of Clifford modules {$GL_{n}(\mathbb{R})/GL_{n}(\mathbb{C})$} check? etc If you multiply all the quotients together than you get the identity (?)

{$\begin{pmatrix} & & \mathbb{C}_{n} & & \\ & \mathbb{H}_{n} & & \mathbb{R}_{n} & \\ \mathbb{H}_{n} \times \mathbb{H}_{n} & & & & \mathbb{R}_{n} \times \mathbb{R}_{n} \\ & \mathbb{H}_{n} & & \mathbb{R}_{n} & \\ & & \mathbb{C}_{n} & & \end{pmatrix}$}

  • Eight-cycle
    • The fourfold periodicity within the Bott periodicity is a repeating pattern (0,1,1,0 ?)
    • W: Symmetric group Note that the reverse on n elements and perfect shuffle on 2n elements have the same sign; these are important to the classification of Clifford algebras, which are 8-periodic.
    • This pattern is based on the sign of the pseudovector.
    • This pattern expresses whether the sum of 1+...+n is even or odd. Even + even = even; + odd = odd; + even = odd; + odd = even; and so on. This pattern is likewise the sum of the numbers mod 2. Thus 0+0=0; +1=1; +0=1;+1=0.
    • This pattern also comes up in multiplying by i. We get 1, i, -1, -i - two positives followed by two negatives.
    • A similar pattern comes up in the volumes of n-spheres. The volume of an n-sphere is {$\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}$} where {$\Gamma(n+\frac{1}{2})=(n-\frac{1}{2})(n-\frac{3}{2})\cdots \frac{1}{2}\cdot\pi^{\frac{1}{2}}$}. Thus we have: {$\frac{2n}{2}=n, \frac{2n+1}{2}-\frac{1}{2}=n, \frac{2n+2}{2}=n+1, \frac{2n+3}{2}-\frac{1}{2}=n+1$}, and so on.
    • The pattern also comes up by two recurrences for volume and hyperarea (surface area) in terms of each other. Starting with {$V_0(R)=1, A_0(R)=2$} and continuing {$V_{n+1}(R)=\frac{R}{n+1}A_{n}(R), A_{n+1}(R)=(2\pi R) V_{n}$} we get the pattern for the factor {$\pi$}.

Real numbers = slack, Complex numbers = null, Quaternions = perspective

Clifford algebra periodicity

  • Readings
  • The Wikipedia article explains that in the complex case the quadratic form there is no signature because {$+1 = i^2(–1)$}. Whereas in the real case the signature distinguishes between the number of {$+1$} and the number of {$–1$}.
  • The pseudoscalar (the product of all the basis elements) is important because it is similar to the scalar. When the dimension is odd, then the pseudoscalar commutes with everything, and so it is in the center. When the dimension is even, then the pseudoscalar does not commute with everything and so the center only consists of the identity.
  • In the real case, what characterizes the Clifford algebra of signature {$(p,q)$} is the number {$p-q \text{mod} 8$}.
  • There are the progressions, adding basis elements which square to {$-1$}, but likewise in the {$+1$} direction:
    • Complexification:

{$$R, C, H$$}

  • Growing:

{$$R, R \bigoplus R, \begin{pmatrix} R & R \\ R & R \end{pmatrix} $$}

  • Untangling:

{$$ H, \begin{pmatrix} C & C \\ C & C \end{pmatrix}, \begin{pmatrix} R & R & R & R \\ R & R & R & R \\ R & R & R & R \\ R & R & R & R \end{pmatrix} $$}

  • Wedderburn's theorem states that The Artin-Wedderburn theorem generalizes this result to Artinian rings.
  • Frobenius's theorem for real division algebras states that every finite-dimensional associative division algebra over the real numbers is isomorphic to the real numbers, the complex numbers or the quaternions.
  • Hurwitz's theorem: The only real normed division algebras are R, C, H, and the (non-associative) algebra O.
  • A division algebra is an algebra over a field in which division, except by zero, is always possible.
  • C0 R
  • C1 C
  • C2 H
  • C3 H + H
  • C4 H(2)
  • C5 C(4)
  • C6 R(8)
  • C7 R(8) + R(8)
  • C8 R(16)

{$C_{n+8}$} consists of 16 x 16 matrices with entries in {$C_n$}.

  • The positive and negative directions (for the signature) are related as {$R$} and {$M_8(R)$}.
  • For any n=s+t (exploring how we generate the perspectives in a division of everything into n perspectives, going from 0 to n in n steps) there are two patterns:
    • even n, {$X=2^{\frac{N}{2}}$}: {$\dots R(2X), R(2X), H(X), H(X), R(2X), R(2X), H(X), H(X) \dots$}
    • odd n, {$X=2^{\frac{N-1}{2}}$}: {$\dots H(X)\bigoplus H(X), C(2X), R(2X)\bigoplus R(2X), C(2X), H(X)\bigoplus H(X), C(2X), R(2X)\bigoplus R(2X) \dots$}
  • Combined, these two patterns yield for n, along the edge:
    • {$\dots R(2^N), C(2^N), H(2^N), H(2^N) \bigoplus H(2^N), H(2^{N+1}), C(2^{N+2}), R(2^{N+3}), R(2^{N+3})\bigoplus R(2^{N+3}) \dots$}

Generalized Clifford Algebra has clock-shift operators.

  • Some matrices describe the 8-cycle clock (the trolley stops).
  • Generalized Pauli matrices describe the 3 shifts (the trolley cars of different increments +1, +2, +3). See the "clock and shift matrices".

Complex structures

  • We call {$J$} a complex structure on {$R^n$} if {$J\in O(n)$} and {$J^2=−I_n$}. Denote the space of complex structures {$Ω_1(n)⊂O(n)$}.
  • Define {$Ω_k(n)$} to be the space of complex structures that anti-commute with fixed {$J_1,\dots ,J_{k-1}$}.
  • {$Ω_0 \cong Ω_8$}
  • Consider how complex structures relate to divisions of everything. Apparently, each {$J_i$} is a perspective. Anti-commutativity {$J_iJ_j = -J_jJ_i$} means that the composition of perspectives is inverted if the order is switched. So the matrix {$-I$} can be interpreted as an inversion of perspective, and thus, of chains of perspectives. A set of eight perspectives brings us back to no perspectives, which is to say, the default perspective at the origin.

Ideas

  • The two-cycle is: the undefined (the nullsome) yields the defined (the foursome), which yields the defined undefined (the eightsome), which is the same as the undefined (the nullsome).
  • The eight-cycle arises from the details: adding four perspectives yields the foursome, the defined. Then we can, dually, think of subtracting perspectives from that. But the subtraction (relaxing, removing perspectives) is defined by adding perspectives, yielding the fivesome which lets a sensor be free in time and space, the sixsome lets the conditional sensor relate to the unconditional, the sevensome lets the external unity (of the whole) match the internal unity (of slack) in a dualistic way, the eightsome has the system collapse.
  • 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
  • Complex case: 2-periodicity - divisions having 4 (nežinojimas) or 2 (žinojimas) representations. Real case: 8-peridocity.
  • 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.

John Baez about Clifford algebra periodicity

Notes

Bott Periodicity

  • 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.
  • How can the fact that {$\Gamma(\frac{1}{2})=\sqrt{\pi}$} relate {$\pi$} and {$e$}? Note that {$\Gamma(1)=0!$} and {$\Gamma(\frac{3}{2})=\frac{\sqrt{\pi}}{2}$}. Consider the integral definition.
  • 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.
  • R nullsome
  • H twosome 1+i (different) j+ij (the same)
  • H+H threesome splits the twosome
  • (H2) foursome - internally doubles
  • (C4) fivesome
  • (R8) sixsome
  • R+R sevensome (dividing the nullsome into two perspectives)

(S16) what would S mean? half of R? positive reals?

Alain Connes about Bott periodicity and CTP. "Why Four Dimensions and the Standard Model Coupled to Gravity...

Attiyah, Bott, Shapiro. Clifford Modules. 1963.

  • Clifford modules are representations of Clifford algebras.
  • They describe the eightfold periodicity.
  • Related to Adams's work on vector fields on spheres.
  • The odd and even dimensional orthogonal groups are related to odd and even dimensional (crosspolytope-hypercube) choice frameworks. How does the duality of center and totality that distinguishes these two choice frameworks add an additional distinction that makes for eightfold Bott periodicity?
  • What is the significance (for Bott periodicity) of {$S^{n-1}=SO(n)/SO(n-1)$} ?

Shintaro Fushida-Hardy. Notes for MATH 282B Homotopy Theory.

  • Chapter 2 on Fibre Sequences contains examples in Sections 2.5 and 2.6.
  • Seems relevant for mastering Bott periodicity. Mentions the fibration for orthogonal groups.

Is Bott periodicity related to the triality of SO(8) ? and thus the unit octonions? See John Baez.

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
  • M. F. Atiyah, R. Bott, and A. Shapiro. “Clifford Modules”
  • So(8) and Bott?
  • Could Feynman diagrams for four fields express the foursome? How goes to What in one direction and Why goes to Whether in the opposite direction. And could Feynman diagrams for N fields express the division of everything into N perspectives? And does Bott periodicity apply?
  • In my understanding of Bott periodicity, is it the + signs that are growing or the - signs?
  • 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.
  • Krulewski bott proof
  • J. F. Adams, Infinite Loop Spaces, Ann. of Math. Studies 90, 1978.
    • 4.66. If F→E→B is a fibration or fiber bundle with E contractible, then there is a weak homotopy equivalence F→ΩB .
    • For each topological group G there is a fiber bundle G→EG→BG with EG contractible, hence by the proposition there is a weak equivalence G ≃ ΩBG. There is also a converse statement
  • Combining this duality between Σ and Ω with the duality between fibers and cofibers, we see a duality relationship between the fibration and cofibration sequences of §4.3:
  • One formulation of Bott periodicity describes the twofold loop space, {$\Omega ^{2}BU$} of {$BU$}. Here, {$\Omega$} is the loop space functor, right adjoint to suspension and left adjoint to the classifying space construction. Bott periodicity states that this double loop space is essentially {$BU$} again.
  • In the corresponding theory for the infinite orthogonal group, O, the space BO is the classifying space for stable real vector bundles. In this case, Bott periodicity states that, for the 8-fold loop space,
  • https://en.m.wikipedia.org/wiki/Periodic_table_of_topological_invariants
  • There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices.
  • Cameron Krulewski. The K-Theoretic Classification of Topological Materials
  • https://en.m.wikipedia.org/wiki/Spin_representation
  • Steven Lehar at PhilPeople
  • Understand in what sense the Clifford algebra for the complex numbers is +1 and -1. We have {$e_ie_j=-e_je_i$}. What is the orientation of {$e_ie_i$}?
  • Understand the Bott periodicity for the Clifford algebras.
  • Interpret Clifford algebras geometrically, what it means for an area to become zero, one or negative one.
  • Representation of Lorentz group and Clifford algebra

Morita equivalence is an additive equivalence

  • Presumably, an additive equivalence is one in which the equivalence is by way of an additive functor.
  • Additive category An additive category is one such that all of its hom-sets are abelian groups, composition of morphisms is bilinear, and it admits all finitary biproducts.
  • Additive functors A functor F: C → D between preadditive categories is additive if it is an abelian group homomorphism on each hom-set in C. If the categories are additive, then a functor is additive if and only if it preserves all biproduct diagrams.
  • Additivity of adjoint functors All adjoint functors between additive categories must be additive functors.
  • If R is Morita equivalent to S, then their centers C(R) and C(S) are isomorphic.
  • Morita equivalence is not interesting in the case of commutative rings because then they are isomorphic.
  • The ring {$M_n(R)$} of n-by-n matrices in R is Morita-equivalent to R for any n>0. Work out the functors here.
  • Morita equivalence could describe a noncontradictory model of contradiction.

Bott periodicity

Complex vector bundles

Keisti - Įkelti - Istorija - Spausdinti - Naujausi keitimai -
Search:
Šis puslapis paskutinį kartą keistas February 23, 2022, at 03:06 PM