Parallel (operator): Difference between revisions

Revision as of 01:56, 23 March 2023

Revision as of 03:00, 23 March 2023

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==Projective view==

Two charts make an [[atlas (topology)|atlas]] that covers the [[complex projective line]]. The first chart covers zero but not infinity: [''z'':1] → ''z'', the second chart covers infinity but not zero: [1:''z''] → ''z''. The projectivities of the projective line are represented by matrices in the [[general linear group]] GL(2,C).:

:<math> \begin{pmatrix} 1 & 0 \\ a & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ b & 1 \end{pmatrix}

:<math>\begin{align}

= \begin{pmatrix} 1 & 0 \\ a+b & 1 \end{pmatrix} </math>

:<math> \begin{pmatrix} 1 & a0 \\ 0a & 1 \end{pmatrix} \begin{pmatrix} 1 & b0 \\ 0b & 1 \end{pmatrix}

&= \begin{pmatrix} 1 & a+b0 \\ 0a+b & 1 \end{pmatrix}</math>,

\\[10mu]

:<math> \begin{pmatrix} 1 & 0a \\ a0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0b \\ b0 & 1 \end{pmatrix}

&= \begin{pmatrix} 1 & 0a+b \\ a+b0 & 1 \end{pmatrix} </math>.

\end{align}</math>

The two matrix products show that there are two subgroups of GL(2,C) isomorphic to (C,+), the additive group of C. Depending on which chart is chosen, one operation is +, the other is <math>\parallel.</math>