Introduction E9F5FC

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Bott periodicity, Bott periodicity models divisions

Linear complex structures

The inverse of an orthogonal matrix is given by its transpose.

{$J_1=\begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix}\equiv i$}

{$J_2=\begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \\ -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{pmatrix}\equiv \begin{pmatrix} 0 & \varphi \\ -\varphi & 0 \\ \end{pmatrix}$}

{$J_1J_2=\begin{pmatrix} i & 0 \\ 0 & i \\ \end{pmatrix}\begin{pmatrix} 0 & \varphi \\ -\varphi & 0 \\ \end{pmatrix} = \begin{pmatrix} 0 & i\varphi \\ -i\varphi & 0 \\ \end{pmatrix}=\alpha$}

{$I_1J_1J_2=\begin{pmatrix} I & 0 & 0 & 0 \\ 0 & I & 0 & 0 \\ 0 & 0 & -I & 0 \\ 0 & 0 & 0 & -I \\ \end{pmatrix}\begin{pmatrix} 0 & i\varphi & 0 & 0 \\ -i\varphi & 0 & 0 & 0 \\ 0 & 0 & 0 & i\varphi \\ 0 & 0 & -i\varphi & 0 \\ \end{pmatrix}=\begin{pmatrix} 0 & i\varphi & 0 & 0 \\ -i\varphi & 0 & 0 & 0 \\ 0 & 0 & 0 & -i\varphi \\ 0 & 0 & i\varphi & 0 \\ \end{pmatrix}$}

{$J_3=(I_1J_1J_2)^{-1}=\begin{pmatrix} 0 & -i\varphi & 0 & 0 \\ i\varphi & 0 & 0 & 0 \\ 0 & 0 & 0 & i\varphi \\ 0 & 0 & -i\varphi & 0 \\ \end{pmatrix}\Rightarrow \begin{pmatrix} 0 & -i\varphi \\ i\varphi & 0 \\ \end{pmatrix}$}

{$LJ_3=\begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ \end{pmatrix}\begin{pmatrix} 0 & -i\varphi & 0 & 0 \\ i\varphi & 0 & 0 & 0 \\ 0 & 0 & 0 & i\varphi \\ 0 & 0 & -i\varphi & 0 \\ \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & i\varphi \\ 0 & 0 & -i\varphi & 0 \\ 0 & -i\varphi & 0 & 0 \\ i\varphi & 0 & 0 & 0 \\ \end{pmatrix}$}

{$J_4=(LJ_3)^{-1}=\begin{pmatrix} 0 & 0 & 0 & -i\varphi \\ 0 & 0 & i\varphi & 0 \\ 0 & i\varphi & 0 & 0 \\ -i\varphi & 0 & 0 & 0 \\ \end{pmatrix}$}

{$J_\gamma=\begin{pmatrix} 0 & -1 \\ 1 & 0 \\ \end{pmatrix}=i$}

{$\begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix}\begin{pmatrix} v_+ \\ v_- \\ \end{pmatrix} = \begin{pmatrix} v_- \\ v_+ \\ \end{pmatrix}$} is the form of an isometry {$L_\beta$} between two vector subspaces.

{$\begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}\begin{pmatrix} v_+ \\ v_- \\ \end{pmatrix} = \begin{pmatrix} v_+ \\ -v_- \\ \end{pmatrix}$} is a nontrivial antilinear operator {$I_\alpha$} that squares to {$+1$}.

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

Suppose we have a linear operator that squares to {$+1$}. What does that look like?