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If you are working in ordinary Euclidean space with orthonormal bases, it doesn't do much for you. When you start doing calculus on embedded surfaces (the begin
by BlackFly 5y ago
If you are working in ordinary Euclidean space with orthonormal bases, it doesn't do much for you. When you start doing calculus on embedded surfaces (the beginning of differential manifolds) it begins to be more helpful. Since Lie algebras are special types of differential manifolds, you can learn quite a bit by studying the geometry and this quickly leads into gauge theory and modern physics. The Geometry of Physics: An Introduction, by Theodore Frankel does a good job illustrating a lot of aspects of geometry even offering some geometric insight into some classical physics.
It is an absolute necessity for general relativity.
- contravariant 5y agoIs there some canonical metric on Lie algebras that makes this possible? Otherwise geometric algebra won't get you far.
- enkimute 5y agoEvery Lie algebra is a bivector algebra (see 'Lie groups as Spin groups' http://geocalc.clas.asu.edu/pdf/LGasSG.pdf http://geocalc.clas.asu.edu/pdf/LGasSG.pdf). Additionally the GA formalism enables closed form solutions for the exponential map for all bivector algebras. (see 'Graded symmetry Groups' https://www.researchgate.net/publication/353116859_Graded_Symmetry_Groups_Plane_and_Simple/ https://www.researchgate.net/publication/353116859_Graded_Sy...).
- Grustaf 5y agoThat books seems very interesting, but is it really about Geometric Algebra? I'm not talking about geometry in general, or differential geometry, which I'm a bit familiar with. And not Algebraic Geometry either, for that matter, which is also a fascinating subject. I mean specifically Geometric Algebra. It sort of seems like a notation, but it has almost a cult like following and perhaps it's more than a notation, is it a theory, a branch of mathematics?
- yuppiemephisto 5y agoIt really is useful even at the advanced level. Following it far enough leads to the Atiyah-Singer Index Theorem and Hodge Theory. The advantage over the exterior algebra is that you have that and an interior algebra, which leads to many formulas in differential geometry becoming very natural (like Cartan's magic formula).
- musgravepeter 5y agoThere is Geometric Algebra for Physicists by Doran and Lasenby (2003). It recasts mechanics, E&M up to gauge theories and GR into geometric algebra. I stalled out at mechanics but I've now taken it off the Tsundoku pile and may give it another chance.
- Grustaf 5y agoThanks for the tip!