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October 20, 2016

Linear Algebraic Groups (Part 2)

Posted by John Baez

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).

  • 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 kk, points are certain 1d subspaces of k 3k^3, and lines are certain 2d subspaces of k 3k^3. In projective geometry we take the symmetry group to be all of GL(3)\mathrm{GL}(3), take points to be all 1d subspaces of k 3k^3, and take lines to be all 2d subspaces of k 3k^3. It thus subsumes Euclidean, elliptic and hyperbolic geometry. In general we define projective nn-space, kP nk\mathrm{P}^n, to be the set of 1d subspaces of k n+1k^{n+1}.
Posted at October 20, 2016 1:07 AM UTC

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