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If I remember correctly, the Hodge star much more closely belongs to geometric algebras than it does to exterior algebras, since you need a nondegenerate biline
by creata 6y ago
If I remember correctly, the Hodge star much more closely belongs to geometric algebras than it does to exterior algebras, since you need a nondegenerate bilinear form to define the Hodge star, from which you can just as easily define the geometric product, and from the geometric product the Hodge star.
To make an analogy, it sounds a bit like you're asking whether inner product spaces and vector spaces are equivalent. Every geometric algebra gives rise to a Hodge star, and an exterior algebra, and so on, but exterior algebras are a much more general concept, so they're less powerful until you tack that extra structure on.
- DreamScatter 6y agoYou are mistaken, the Hodge star does not belong to geometric algebra more than exterior algebra, that's the wrong way to look at it. Exterior algebra is just only a sub algebra of geometric algebra, they both have the same Hodge star. Saying the Hodge star belongs more in one than the other is a bit silly.
- creata 6y agoLook, you already need a bilinear form to get the Hodge star. My point is that with that same bilinear form you also get an entire Clifford algebra, and a much more natural definition of the Hodge star. That's all I'm saying. Is that mistaken?
- yiyus 6y agoTo be honest, my question is much more practical, and much naiver too. I work with people who use Euler angles to express rotations, and it's a horrible world. I learned quaternions in my day, and there are some obvious advantages. When I discovered GA some years ago, it was really eye opening. It makes quaternions an easy to explain concept, even intuitive, and I've used it since then, not only in my own work but also to teach other people. Then, I learned about differential forms, and it's also very interesting, I think I could base my "intuitive explanations" in this other paradigm, but I'm not sure I should. I do not think it makes a big difference in my particular case, but as I said I find this "competition" a bit frustrating, and am trying to understand it better. I cannot discuss with a mathematician if the Hodge star is a GA or a DFs concept, but I have found it all over the place when reading about DFs, and not so in GA related material (though I have a vague idea about how the Hodge star operator can easily be defined in GAs using the pseudoscalar). But is this really my choice? I have listened opinions about which one is more general, but not really convincing arguments (at least not arguments that are obvious to me). Thanks for your explanation. I think I need to have a deeper look at this stuff. I like to get lost in these rabbit holes, but sometimes it goes a bit over my head.