Talk:Geometric algebra

message: Hi HongGong, in https://en.wikipedia.org/w/index.php?title=Geometric_algebra&diff=1016190290&oldid=1016055619 (reverted) and then in https://en.wikipedia.org/w/index.php?title=Geometric_algebra&diff=1030284754&oldid=1018780264 you changed the notation for reversion from dagger to tilde. I searched the talk page for an explanation, but could not find any. Your second edit comment merely says "The notation is correct." Notation is a convention. Since the dagger notation was defined in the article itself (just as the tilde notation now is), it was just as correct as the tilde notation (as long as it does not clash with other notation). I noticed that you've also changed a different article in a similar way (https://en.wikipedia.org/w/index.php?title=Rotor_(mathematics)&diff=1016188882&oldid=1001315838), with an edit comment of similar quality ("General cleanup"). Can you please explain what makes tilde better than dagger here? --RainerBlome (talk) 21:49, 3 December 2021 (UTC)
thread_title:  Reversion notation 

message: In the article the Grassmann exterior product is - be it between quotes - referred to as the "meet". Allthough both concepts use the same symbol, in my opinion there is no relation between the two. Madyno (talk) 16:46, 2 December 2022 (UTC)
replies: text: It says that the meet is the dual of the Grassmann exterior product. -Bryan Rutherford (talk) 17:44, 2 December 2022 (UTC)
thread_title:  Meet 

message: I undid a change by , relating to defining the algebra as what Vaz&daRocha and Lounesto appear to term "universal".  I'm not objecting to the idea that the properties as stated do not necessarily define a universal Clifford algebra in this sense (and we do want the universal case), but we should provide a clearer definition, rather than some wording that someone may only understand if they are familiar with universal properties.  Lounesto does this by requiring that Cl(V, g) is not generated by any proper subspace of V, while it is by the whole of V (Lounesto 2001 Clifford Algebras and Spinors p. 190).  This seems to be more understandable to a reader.  One possible shortfall in this is that he does not seem to cover the case of a degenerate quadratic form in this claim (though I suspect that it still applies).  Vaz&al prove that requiring that dim(Cl(V, g)) = 2dim V is sufficient for it to be universal (Vaz & da Rocha 2016 p. 58).  Maybe we can use these ideas?  —Quondum 14:46, 2 March 2024 (UTC)
replies: text: My point with those edits is that just giving those properties is not a definition in itself. I'm perfectly fine if there's a better yet still elementary way to word it because most casual readers won't understand abstract algebra here. I suggest adding the "not generated by a proper subspace but by the whole space condition" and the sentence "It can be shown that these conditions uniquely characterize the geometric product".--Jasper Deng (talk) 19:38, 2 March 2024 (UTC)
replies: text:  I'll spend a bit of time on it.  I'll model it on your suggestion initially, which I like.  I might include footnotes that may need trimming/rework.  —Quondum 21:33, 2 March 2024 (UTC)
replies: text:  I've tried to do this.  It turned out that the condition that I gave above was sufficient only in the case that the quadratic form is nondegenerate, which is rather unsatisfying: some useful GAs have degenerate quadratic forms.  To keep the less abstract part uncomplicated, I've chosen to not call it "axioms", only "properties", and hence not needing them to definitively define the geometric product.  The result is a bit clumsy; feel free to refine.  —Quondum 01:05, 3 March 2024 (UTC)
thread_title:  Universal/general algebra 

message: The lead currently contains a statement "... pseudovector quantities of vector calculus normally defined using a cross product, such as ... the magnetic field".  I am not familiar with the magnetic field "normally" being defined in this way, although I'm aware if the Biot–Savart law, which is not only a cross product.  The point is that few readers will find this helpful; examples such as this are meant to be familiar.  Maybe this particular example (magnetic field) might be better removed?  —Quondum 02:04, 13 March 2024 (UTC)
thread_title:  Unhelpful example?