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Modeling introspection, Spin representations, ND

Andrius Kulikauskas: I am overviewing

Bott periodicity interpretations

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Centered on {$\mathbb{R}\oplus\mathbb{R}$}

Mappings

The Chevalley action is from the number of perspectives on the number of shifts in perspective.

We are working with complex Clifford algebras and the generators square to {$+1$}.

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Centered on {$\mathbb{R}$}, with 8 generators {$e_j^2=-1$}

Eighth roots of unity {$e^{-\frac{\pi}{2}i}$}

{$e^{-\frac{\pi}{2}i}=\frac{\sqrt{2}}{2}(1-i)$}

{$(1-i)^k$}

Linear complex structures

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Centered on {$\mathbb{R}$}, with 8 generators {$e_j^2=1$}

Eighth roots of unity {$e^{\frac{\pi}{2}i}$}

{$e^{-\frac{\pi}{2}i}=\frac{\sqrt{2}}{2}(1+i)$}

{$(1+i)^k$}

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Centered on {$\mathbb{R}$}, with 8 generators {$e_{\pm j}^2=\pm1$}

Super division algebras

Automorphisms

Shifts in perspective

Consider three shifts in perspective: {$J_1J_2$}, {$J_3J_4}, {$J_5J_6$}

First shift in perspective

{$J_1J_2=ij=k$} is a pseudoscalar.

Second shift in perspective

{$J_3J_4=L$} is the isometry. It is central in the interpretation in terms of the octonions. It is the looking glass.

Third shift in perspective

{$J_5J_6$}

Universal

John Harland: Bott periodicity and recursion * Thue-Morse sequence

{$K_1=1$}

{$J_{j-1}= \begin{pmatrix} 0 & -K_{j-1}^T \\ K_{j-1} & 0 \\ \end{pmatrix}$}

{$K_j=\begin{pmatrix} K_{j-1} & -0 \\ 0 & -K_{j-1}\\ \end{pmatrix}$}

{$k_1=1$}

{$k_j=k_{j-1},-k_{j-1}$}