Linear Algebraic Groups (Part 2)

October 20, 2016

Posted by John Baez

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This time we show how projective geometry ‘subsumes’ Euclidean, elliptic and hyperbolic geometry. It does so in two ways: the projective plane includes all 3 other planes, and its symmetry group contains their symmetry groups.

By the time we understand this, we’re almost ready to think about geometry as a subject that depends on a choice of group. But we’re also getting ready to think about algebraic geometry (for example, projective varieties).

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Lecture 2 (Sept. 27) - The road to projective geometry. Treating Euclidean, elliptic and hyperbolic geometry on an equal footing: in each case the symmetry group is a linear algebraic group of 3 × 3 matrices over a field $k$ , points are certain 1d subspaces of $k^3$ , and lines are certain 2d subspaces of $k^3$ . In projective geometry we take the symmetry group to be all of $\mathrm{GL}(3)$ , take points to be all 1d subspaces of $k^3$ , and take lines to be all 2d subspaces of $k^3$ . It thus subsumes Euclidean, elliptic and hyperbolic geometry. In general we define projective $n$ -space, $k\mathrm{P}^n$ , to be the set of 1d subspaces of $k^{n+1}$ .

Posted at October 20, 2016 1:07 AM UTC

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October 20, 2016