Math 4 Wisdom. "Mathematics for Wisdom" by Andrius Kulikauskas.

I want to overview my results and my research how Bott periodicity models introspection

Overviews: Modeling introspection, Bott periodicity flavors, Bott periodicity models divisions, Divisions
Expositions: Investigating Bott periodicity, Modeling Introspected Contexts With Mutually Anticommuting Linear Complex Structures, Bott periodicity for Clifford algebra maniacs, Bott periodicity for octonion maniacs, Artificial general intelligence presentation, Modeling the self
Key topics: Hamiltonians, Quantum symmetries, CT groups, Antiautomorphisms, Lie group embeddings, Lie algebra decomposition, Super division algebras, Super Brauer Group, Clifford algebras, Symmetric spaces, Octonions, Binomial Bott, KTheory, Dynkin diagrams, Spinors, Random matrices, Krebs cycle, NSpheres, Linear complex structures
Related topics: Symmetry fibrations, Topological invariants, Homotopy groups, String theory, 24 Cell

An Allegory: The Solipsistic Self as the Hamiltonian of a Noninteracting Fermion

Bott Periodicity Models Consciousness? Preliminary Exploration

Modeling Introspection

Understand how Moore's table of CT groups aligns with Stone-Chiu-Roy's table of symmetric spaces.
- Understand how Moore relates CT group extension {$(\phi,\chi)$}-representations and super division algebra graded representations.
Consider how time reversal (forwards to backwards) can be identified with space reversal (positive to negative) if we add an extra dimension.
Consider how the quantum symmetry {$T$} or {$C$} squaring to {$+1$} or {$-1$} expressing unconsciousness and consciousness, and how that interacts with the quantum symmetry, as in "consciously inaccessible".
Relate the Clifford algebra recursions with the eighth root of unity.
Relate Dynkin diagrams of {$SO(n)$} and sequences of shifts in perspectives.

Goals

I wish to model

The structure of each division of everything
- The perspectives, the shifts in perspective, and their relationships.
Why the division of everything into eight perspectives collapses into a division of everything into zero perspectives.
How the three minds act as operators adding a perspective, a perspective on a perspective, and a perspective on a perspective on a perspective.
The 24 equations which result from the 3 minds acting on the 8 divisions of everything.
The 6 conceptions by which we conceive the 8 divisions of everything.
The 12 circumstances isolating individual perspectives.

With regard to this model, I want to make sense of various mathematical phenomenon.

The maps of Clifford algebras: reversing, conjugating, involution.
The Clifford algebras given by the {$CT$}-groups and the generators {$C, iC, iCT$}.
The homotopy groups of {$O(\infty)$} and {$U(\infty)$}.

Models

The Structure of a Division of Everything

The Chevalley action explains how an ordered set of generators of {$W$} (an ordered set of shifts in perspective) is acted upon by a basis element of {$V$} (a perspective), or a bivector - a generator of the Lie algebra for the spin group, or a monomial of basis elements.

This has us think of a division of everything as an ordered set of shifts in perspectives with perhaps an additional perspective left over.

Dynkin diagrams for {$SO(n)$}

A shift in perspective is a simple root.
Shifts in perspective are related by 120 degree angles, except for the 135 degree angle indicated in {$B_k$}
An additional perspective makes for {$B_k$} rather than {$D_k$}.
{$SO(8)$} with four shifts in perspective is the same as {$SO(0)$} with zero shifts.
{$SO(1)$} has zero shifts in perspective and an extra long angle that we don't see.
{$SO(2)$} has one shift in perspective but we don't see it.
{$SO(3)$} has one shift in perspective (which we see: the simple root {$x_1$}) and an extra long angle (which we don't)

Perspectives

We can identify perspectives with linear complex structures {$J_k$}.

These {$J_k$} anticommute, which means that they are fermionic, as perspectives should be. They are all distinct. {$J_aJ_b=-J_bJ_a$} implies that if {$J_a=J_b$} then {$J_a^2=0$}, and if {$J_a$} is invertible, then {$J_a=0$}.

The First and Last Perspectives

The first perspective {$J_1$} functions as {$i$}.

The last perspective {$J_k$} is related to the Hamiltonian {$H=im=J_1m$}. {$m$} commutes with {$J_1,J_2,\cdots, J_{k-1}$} and anti-commutes with {$J_k$}.

Shifts in Perspective

The Clifford algebra {$Cl_{0,2}$} and likewise {$Cl_{2,0}$} can be identified with a shift-in-perspective {$e_1e_2$} and its two perspectives {$e_1,e_2$}.

A Clifford algebra {$Cl_{0,n}$} breaks down (as a tensor product) into a shift-in-perspective {$Cl_{0,2}$} tensored with the Clifford algebra {$Cl_{n-2,0$} whose generators square to the opposite sign. The isomorphism is given by Dekter Chua and also Figueroa.

{$e'_1\rightarrow 1\otimes e'_1$}
{$e'_2\rightarrow 1\otimes e'_2$}
{$e'_i\rightarrow e_{i-2}\otimes e'_1e'_2$}

The Three Minds

The three minds are given by three interpretations for a Hamiltonian {$H$}.

It can be interpreted as the transpose {$H=G^T$}. This is the first mind. It corresponds to {$ABC\rightarrow CBA$} and time reversal.
It can be interpreted as the conjugate {$H=G^*$}. This is the second mind. It corresponds to {$ABC\rightarrow (-A)(-B)(-C)$} and charge conjugation.
It can be interpreted as self-adjoint {$H=H^\dagger$}. This is the third mind. It corresponds to {$ABC\rightarrow (-A)(-B)(-C)$} and mirror symmetry.

The three minds are given by quantum symmetries and related Clifford algebra mappings.

Unconscious	Accessible	Time reversal	usual antilinear	{$U_TH^*U_T^\dagger=H$}	Reversing antiautomorphism	{$ABC\rightarrow CBA$}
Conscious	Inaccessible	Charge conjugation, particle-hole	transposing linear	{$U_CH^*U_C^\dagger=-H$}	Conjugation antiautomorphism	{$ABC\rightarrow (-C)(-B)(-A)$}
Consciousness	Definite	Parity, space reversal, sublattice symmetry	transposing antilinear	{$U_SHU_S^\dagger=-H$}	Involution automorphism	{$ABC\rightarrow (-A)(-B)(-C)$}

We can translate various concepts between quantum symmetries and the Clifford algebra mappings

{$H^*=H^T$} conjugation or transpose	reversion {$ABC\rightarrow CBA$}
{$-H$} transposing	reflection of generators
{$H$} antilinear	?

Study Varlamov 2001, Varlamov 2004.

Adjoint operators: Note that the null space of an operator {$T$} equals the complement to the range of its adjoint {$T^*$}. This means that we can describe the same space as a null space mapping to zero (algebraically) and as that which is outside of the range, which doesn't get mapped to (logically). And when we have a self-adjoint operator the null space must be simply {$0$}.

Adjoints are like complex conjugates. A self-adjoint operator is like a real number.

Mathematical phenomena

CT groups

Gregory Moore. Quantum Symmetries and Compatible Hamiltonians., Section 16, page 138, has a table of CT groups and related Clifford algebras. There is an equivalence of categories between the {$(\phi,\chi)$}-representation of the {$CT$}-group and the graded representation of a Clifford algebra. Section 11, page 77, defines {$(\phi,\chi)$}-representation of {$G$}.

Each element of {$G$} is mapped to a real-valued matrix that is either {$\mathbb{C}$}-linear or {$\mathbb{C}$}-anti-linear, as given by {$\phi$}, and either even or odd, as given by {$\chi$}.

The {$\phi$}-twisted extension of {$G$} is relating the structure of {$G$} (which is like two switches, as with the foursome) with recurring activity, the circle {$U(1)$}.