Talk:Exterior algebra

Style changes

@Kavuldra: In https://en.wikipedia.org/w/index.php?title=Exterior_algebra&curid=221537&diff=1116584177&oldid=1116422957#Formal_definitions_and_algebraic_properties, Kavuldra cleaned up the presentation of some mathematical text. While reviewing his improvements I noticed So,

${\textstyle \bigwedge }(V)=T(V)/I$

is an associative algebra. and wonder whether it would read better inline as So, ${\textstyle {\bigwedge }(V)=T(V)/I}$ is an associative algebra. Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:25, 19 October 2022 (UTC)

Presumably the goal is to call this out dramatically, since this is being used as a definition. –jacobolus (t) 20:34, 19 October 2022 (UTC)

Clarifying the relation between the two ways to see exterior algebra: first, as a quotient algebra and second, as the space of antisymmetric multivectors

I have added details the better understand the relation between the two approaches, this being mostly important for positive characteristic of the field. This is pretty much connected to the two possible products given below in the text. 0ctavte0 (talk) 16:04, 12 June 2023 (UTC)

Vector area is special case

@Onel5969: Edit https://en.wikipedia.org/wiki/Talk:Exterior_algebra changed [[oriented area]] to [[Vector area]oriented area] in order to disambiguate it. However, "vector area" only exists in 3 dimensions. I'm not sure how best to correct it while keeping it concise. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:17, 6 August 2023 (UTC)

Too long

This article is way too long, which is especially a problem in the introduction and the first sections. It is unconcise to the point of being unclear, and left me with a fear of wedge products for ages when I first read the article years ago. It is possible to write an intro that's accessible both to physicists and mathematicians while cutting down on the length a lot. If people don't mind I'll give it a thorough edit. 27 Sep 2023 - MM — Preceding unsigned comment added by MeowMathematics (talk • contribs) 13:43, 27 September 2023 (UTC)

This article is not too long (indeed, it should probably be expanded). But the lead section is too long, and the article would benefit from having the first few sections rewritten with an eye toward making them accessible to nontechnical readers. –jacobolus (t) 14:05, 27 September 2023 (UTC)

Shortening it while keeping it clear and accurate is a challenge. If you can write an intro that's accurate, accessible both to physicists and mathematicians, and shorter, I'd encourage you to do so. You could either discuss the proposed lead here or WP:BEBOLD -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:14, 27 September 2023 (UTC)

Thanks for the comments. I propose the using the introduction to walk through what $\wedge ^{n}V$ is, with n = 1, 2, 3 as explicit examples, and with the guiding example of water flowing in R^k, and using the examples when k = 2 of the flow vector field and the curl as examples of n = 1, 2, and in dimension k = 3 the flow vector field, curl and divergence for n = 1, 2, 3. Since in school we introduce vectors using velocity, i.e. as tangent vectors to flow lines, it makes sense to introduce higher wedge products as curls and divergences of vector fields. You could then use all of this to motivate the conditions you want the formal objects $v_{1}\wedge \cdots \wedge v_{n}$ to satisfy in the definition section below. I think I could also draw some nice pictures for this, which would be less confusing than the current first picture, which I feel is a bit notation overloaded for a lead picture. Probably one could also include pictures of EM fields later in the page too and write about how the magnetic field is a 2-form, etc. . -MM — Preceding unsigned comment added by MeowMathematics (talk • contribs) 21:16, 29 September 2023 (UTC)

it makes sense to introduce higher wedge products as curls and divergences of vector fields I think it makes most sense to introduce wedge products as the signed hypervolumes of parallelepipeds, which is elementary geometry/linear algebra, and leave more advanced interpretations (related to multivariable calculus) for later sections. –jacobolus (t) 01:07, 30 September 2023 (UTC)

Were you personally introduced to vectors as being "signed lengths of lines"? I feel like the definition as derivative of position is much easier to understand, and not more advanced. Fwiw, I am not proposing giving the formulas definition of div and curl. 130.226.87.14 (talk) 16:43, 30 September 2023 (UTC)

A (physics-style) vector is a quantity which has a linear direction and a magnitude. Anything involving derivatives is a much more sophisticated and subtle concept. –jacobolus (t) 17:07, 30 September 2023 (UTC)

I was originally introduced to vectors that way, but I later learned just how misleading it is, not just in Mathematics but also in Physics. It's arguably only one of many examples of a vector, and far from the most important. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:02, 1 October 2023 (UTC)

Can you be explicit about which definition you are talking about? Mathematicians have later defined "vector" non-geometrically as any kind of object that can be added toegether or multiplied by scalars in an arbitrary field. But in my opinion they should have picked a different word for this. The concept of a quantity that has a linear direction and a magnitude is too important to not have a dedicated name. –jacobolus (t) 15:35, 1 October 2023 (UTC)

You've already summarized my answer: a vector is an element of a vector space over some field. Of most importance to physics are vector spaces over $\mathbb {C}$ or $\mathbb {R}$.

If you want a name for a vector in $TM$ or $T^{*}M$, directed segment comes to mind, although that isn't general enough. But isn't momentum more important than velocity in contemporary physics? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:24, 2 October 2023 (UTC)

It's more of a motivating heuristic than definition, but I agree that it should stay. One issue is that it's not actually correct, e.g. what is the 2-tensor $e_{1}\wedge e_{2}+e_{3}\wedge e_{4}$ the hypervolume of? It's not, so if you lean too strongly into this angle you'll leave people thinking only wedge products of the form $v\wedge w$ exist. MeowMathematics (talk) 17:07, 2 October 2023 (UTC)

$e_{1}\wedge e_{2}+e_{3}\wedge e_{4}$ is not a wedge product ("blade"), but the sum of two separate parts, each of which can certainly be thought of as an oriented area. –jacobolus (t) 18:29, 2 October 2023 (UTC)

Since it's the only complaint people have had so far, I'll give a sketch of (one of) the ways I propose introducing $V\wedge V$, as an example, and if people are satisfied with it I'll have a go at editing the article soon. Assume the local writing quality will improve, this is just a sketch, and things will be clearer with pictures.

[draw a picture of fluid flow lines, with a number of tangent vectors $v$ drawn, and also squares with $\circlearrowright$ drawn on and arrows along two edges $u,w$ drawn on to demonstrate curl]

In the flow of a fluid in two dimensions, the velocity of the fluid at a point is a vector $v\in V$. It measures how fast a fluid particle is moving and in what direction. To measure it, draw a small line $\ell$ through the point and measure how many particles pass through it a second. The velocity vector is uniquely determined by (this for all line elements). [note: maybe give a formula for this, or probably better just pictorally explain it in the diagram below]

[picture of line elements at random angles at the points where we drew tangent vectors, and points where we drew the circles]

We can instead measure how much the fluid is rotating around a point $p$ [note: same note as above], which is called the curl of the fluid. We represent it with a rectangle at $p$ with area proportional to the curl. If the rectangle has edges $u,w\in V$ let us denote the curl by $u\wedge w$. Since we want the curl to only depend on the area, we want the curl to satisfy:

[draw pictures justifying $(u+u')\wedge w=u\wedge w+u'\wedge w$, and the other wedge identities, in the 2d case]

[Then do the same thing in 3d. Vectors correspond to flow through a 2d plane, curl flow around a 1d line, divergence flow in to a point. Explain how these are elements of $V,V\wedge V,V\wedge V\wedge V$. Conclude that an element of $V\wedge V$ gives you a number for every 2-plane, thus can be written in basis form as a sum of $e_{i}\wedge e_{j}$]

[It might be less confusing to introduce the 3d case in parallel, since $\wedge ^{2}\mathbf {R} ^{2}\simeq \mathbf {R}$ might confuse people?] MeowMathematics (talk) 17:02, 2 October 2023 (UTC)

This seems like a good example to put about halfway down the page, but does not seem like a good introduction. You might also consider working on curl. –jacobolus (t) 18:34, 2 October 2023 (UTC)

OK, take 2 on an alternative introduction.

[current first 2 sentences]

[picture in 3d: 1) different multiples of the three basis vectors v_i and their sum, marked v = v₁ + v₂ + v₃. 2) different multiples of the three basis planes Π_i of ∧²R³ and Π = Π₁ + Π₂ + Π₃. 3) a three-cube C.]

The first example of an exterior algebra is a vector space V, with addition law [draw picture indicating vector addition and multiplication by constants]

The next example is its second exterior power ∧²V, which is the vector space spanned by plane elements Π with addition law [draw picture indicating plane addition (i.e. (u+u')∧w = u∧w + u'∧w, but include this formula in this picture) and RHS version) and multiplication by constants]. Each plane is determined by its edges, so we write Π = u∧w. The (above picture) induces [(u+u')∧w = u∧w + u'∧w, etc.]. Elements of ∧²V are called 2-vectors.

[repeat this explanation for n = 3 and n = 0]

A) What precisely is the first projective geometry remark trying to say? That you can take P(∧ⁿV)? If it's just that we should cut it.

B) The Plucker coordinate part should be rewritten, something like "wedge powers allow us to define the Grassmannian: the space of k-planes in Rⁿ. It is precisely the subspace ofk-multivectors in P(∧^kV) which are of the form u₁ ∧ ... ∧ u_n. This is cut out by quadratic equations, called Plucker coordinates."

C) The paragraph on rank, kernel etc belongs in the definition section below.

The exterior algebra is k + V + ∧²V + ..., and has a product

[The formula u . (v∧w) = u∧v∧w and a picture depicting it, and maybe one or two more examples]

making into an associative algebra. This is a lot like homogeneous polynomials of degree

, such that when elements of different degrees are multiplied, the degrees add for the degree of the product.

D) The penultimate paragraph should be cut or moved to the definition section.

E) Trim the last paragraph. Keep that the definition works for any module, and the relation to differential forms. Maybe mention other relations, e.g. that physicists call wedge forms "ghosts". MeowMathematics (talk) 00:01, 3 October 2023 (UTC)

I find that text confusing, especially the term plane element, and it is not clear when you are talking about a k-blade and when you are talking about an arbitrary element of $\wedge ^{k}V$

I also don't understand what you mean by Plücker coordinates being cut out.

I'm aware of physicists using the term ghosts in Quantum Mechanics, but I'm not aware of any other use, and certainly not as an equivalent of differential forms. Do you have an example of such a use? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:07, 3 October 2023 (UTC)

The definition of $\wedge ^{2}V$ I was trying to make in low-tech language is: it's spanned by oriented 2-parallelpipeds inside V with base at the origin (which you can identity with rank 1 tensors in $\otimes ^{2}V$) modulo [relations]. I didn't want to say "oriented 2-parallelpipeds inside V with base the origin", but I'm not sure what to call this (n.b. k-blades/multivectors live in the quotient by [relations] so that's not quite the right word either). Maybe "plane element" is bad, do you have other suggestions? Just "2-planes"? Combined with the picture of an oriented 2-parallelpipeds inside V with base the origin that's probably pretty unambiguous right?

I mean ${\text{Gr}}(k,n)$ is cut out by Plucker coordinates in $\mathbf {P} (\wedge ^{k}V)$, which are quadratic.

This happens in BRST quantization, or more generally whenever the hilbert space of states is actually a complex (>0 degree part is called ghosts).

Though actually, maybe we should just make a remark after differential forms about the Koszul complex, and in the article on that say something about ghosts.

MeowMathematics (talk) 13:34, 3 October 2023 (UTC)

$\wedge ^{2}V$ is isomorphic to the antisymmetric subspace of $\otimes ^{2}V$; using quotient spaces seems to be an unnecessary complication.

I would use oriented area, but there may be a clearer term.

It's cut out that I'm having trouble with. Are you trying to say that Plücker Coordinates satisfy a quadratic relation?

The ghosts in BRST quantization are in an anticommuting algebra that is not the exterior algebra of a vector space. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:22, 3 October 2023 (UTC)

If you need a target, you can link the term 'oriented area' to bivector. –jacobolus (t) 16:29, 3 October 2023 (UTC)

I disagree with the first point. n.b. the quotient definition is morally closest to what is currently in the introduction ("generated by symbols u∧v which satisfy u∧v = - v∧u"), since if someone tells you to axiomatise the first clause "space generated by symbols u(?)v" where (?) is a symbol you don't know yet, the most natural first guess would be $\otimes ^{2}V$.

In any case, we don't have to use the word quotient space, I think describing the vector space in terms of generators and relations is fine (and this is agnostic to whether you're working with the quotient or subspace definition).

Yeah, oriented area and linking to bivector works pretty well, let's go with that.

Sorry, I'm just saying that ${\text{Gr}}(k,n)\subseteq \mathbf {P} (\wedge ^{k}V)$ is a closed subspace, cut out by (= defined by the zero sets of) certain quadratics $q\in {\text{Sym}}^{2}(\wedge ^{k}V)^{*}$, iirc. MeowMathematics (talk) 17:55, 3 October 2023 (UTC)

In my opinion the lead section of this article (or better the first few sections) should be as non-technical and jargon free as possible, ideally accessible to a high school audience, or at least to a student who has gone through an introductory one-semester linear algebra course. Anything about tensors, quotient spaces, generators, etc. should be skipped until later. Stuff like "abstract skew(anti)-commuting objects", "basic field is supposed of 0 or odd characteristic", "somehow similar to homogeneous polynomials, just being skew-commutative or anticommutative;" etc. is right out. I think what happened here was that multiple articles were merged together and then various extraneous material accreted in the lead section, until we wound up with the current mess. –jacobolus (t) 18:16, 3 October 2023 (UTC)

I definitely agree. However, I get the impression there's some sort of communication error. The text of my past two comments is not what I'm proposing putting into the definition, my "take 2" is. My past 2 comments was just intended to help people here (who know about tensors, quotients, ... ) understand what my sketch in the "take 2" is saying secretly, but the proposed introduction is self-contained and understandable without understanding any of these concepts.

Ah right I see. That must have been years ago, it's been like this as long as I can remember, so it's definitely overdue a rewrite.

Also, I don't know wikipedia's norms very well. At what point can I assume a consensus has been reached and try editing the introduction? MeowMathematics (talk) 18:41, 3 October 2023 (UTC)

The context of my comments was your reference to tensors in $\otimes ^{2}V$) modulo [relations], where taking a quotient space rather than a subspace seems to be an unnecessary complication. If you want to start with formal sums of $u\wedge v$ terms then taking the quotient is appropriate.

I find ${\text{Gr}}(k,n)\subseteq \mathbf {P} (\wedge ^{k}V)$ is a closed subspace, defined by the zero sets of) certain quadratics $q\in {\text{Sym}}^{2}(\wedge ^{k}V)^{*}$ to be a lot clearer than cut out by.

I'm still learning wikinorms myself )-: My advice is to do a draft article, moving as much out of the lead as you can, and distinguishing between definitions and motivations. I'd like to see the lede mention that it is an abstraction of ... and to concentrate on finite dimensional vector spaces over $\mathbb {R}$ without implying that it is the only case. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:52, 4 October 2023 (UTC)

I have boldy rewritten the lead. My changes consist mainly in removing repetitions, philosophical considerations, vague formulations and some sentences that are too technical here, such as the mention of category theory. Also I have added a short definition of the exterior algebra. A large part of the content that I have removed has been kept as blind comments, for an easy use for later edits.

I believe that my edit is a convenient starting point toward a decent version of the article. However, I have not really read the remainder of the aticle nor verified whether the new lead contains a summary of the body. So, probably some paragraphs must be added for this purpose, and, in compensation, some technical details should be moved to the body. In any case, this will be much easier with the new version than with the previous one. — Preceding unsigned comment added by D.Lazard (talk • contribs) 17:43, 4 October 2023 (UTC)

D.Lazard Your version of the lead section is significantly better than before, thanks. I wonder if we can still make it less technical somehow. I'm worried that this might still exclude some less technical readers who could otherwise benefit from understanding what the wedge product is. –jacobolus (t) 17:45, 4 October 2023 (UTC)

Sorry for forgetting to sign: when using the "reply" button, the signature is automatically provided, and here I did not use this button.

Making the lead less technical is a challenge, and I have no idea for that. D.Lazard (talk) 17:53, 4 October 2023 (UTC)

(Feel free to replace the unsign template with a proper signature.) For example, I wonder if there's a way to defer "Moreover, the exterior algebra is universal in the sense that, for every algebra A that has these properties, there is a unique algebra homomorphism from the exterior algebra to V that fixes A" to later in the article instead of putting it right at the beginning. I would expect that the technical jargon "universal", "unique algebra homomorphism", "fixes" might be inaccessible to anyone without a pretty extensive undergraduate pure math background. –jacobolus (t) 18:00, 4 October 2023 (UTC)

Okay, I did some rearrangement. Does that still seem okay? –jacobolus (t) 18:41, 4 October 2023 (UTC)

This is OK. On my side I have made less technical the description of the universal property. I leave to you to decide whether the new version (or a variant) is worth to be restored in the first paragraph. Personally, I think this would be useful, since this provides an accurate definition (up to an isomorphism) of the exterior algebra at a very low level of technicity. D.Lazard (talk) 20:59, 4 October 2023 (UTC)

Some other things still possibly worth mentioning in the lead include that the algebra is alternating, and that, when given an additional metrical structure, multivectors are also the objects of a geometric algebra (Clifford algebra). It may also be worth mentioning the cross product and determinants. –jacobolus (t) 21:08, 4 October 2023 (UTC)

I agree that many other things should be mentioned in the lead. But the lead is already long enough, and this would need to move some content of the present lead to the body. Indeed, a large part of the present lead could be moved in a definition section, and replaced with more on the context, the applications, and the generalizations (such as geometric algebra). This cannot be done reasonably with the present structure of the article. So, I'll open a new thread for discussing this structure, especially the sections on motivations and definitions. D.Lazard (talk) 09:15, 6 October 2023 (UTC)

Thanks! I'll try to edit this a little bit in line with the previous discussion above. Edit: complete. MeowMathematics (talk) 12:12, 7 October 2023 (UTC)

The edit introduces the term oriented length into the lead; lots of vectors occur in Physics that are not oriented lengths. The term is appropriate in the examples section, but not as part of a definition. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 04:32, 8 October 2023 (UTC)

To follow this, I think "directed magnitude" or "oriented magnitude" is better than "oriented length". The latter implies a Euclidean vector space, but this is an affine notion, without a need for any universal definition of length. –jacobolus (t) 01:56, 10 October 2023 (UTC)

Structure of the first sections.

The first section after the lead is § Motivating examples. This section is a mess, as it supposes already known what is the wedge product, and it uses many other concepts that are not supposed to be known by most readers, such as "a metric tensor field and an orientation", "the orientation $\mathbf {e} _{1}\times \mathbf {e} _{2}$ and with the metric ${\begin{bmatrix}1&0\\0&1\end{bmatrix}}$", etc. So, this section must be moved after a definition section, shortened, and rewritten for showing that use of exterior algebra simplifies these examples. Such a renewed section could also be renamed "Motivating applications".

The next section is § Formal definitions and algebraic properties. In particular, it uses "is defined" for one of several definitions, and is uses a definition that involves too much algebraic knowledge. So this section must be moved later in the article, renamed "Other definitions", include the definition with the universal property, and show that the definition as a quotient of the tensor algebra is, in fact, a proof of the existence of a solution for the universal property.

This implies to create a first section "Definitions" after the lead that could begin with Let V be a vector space over a field F, and $(e_{i}\mid i\in I)$ be an ordered basis of it. For every natural number, the kth exterior power of V is the vector space ${\textstyle \bigwedge ^{k}V}$ that has a basis formed by the "formal wedge products" $e_{i_{1}}\wedge \cdots \wedge e_{i_{k}}$ such that $i_{1}<\cdots <i_{k}.$ Using the convention that the empty product is 1, one has ${\textstyle \bigwedge ^{0}V=F}$ and ${\textstyle \bigwedge ^{1}V=V.}$ The exterior algebra $\bigwedge V$ is the direct sum ${\textstyle \bigoplus _{k=0}^{\infty }\bigwedge ^{k}V,}$ that is, the vector space with all above formal wedge products as a basis. Then, the section may continue with the definition of the (wedge) product in this vector space, the fact that this makes the exterior algebra an alternating algebra, the fact that a change of basis for V induces a change of basis on the exterior algebra, and thus that the definition does not really depend on the base choice, etc. This in this section that the definitions of blades, multivectors, etc., given in the lead could be moved.

I'll probably not have the time for restructuring the article myself. So, if there is a consensus for the above project, it would be great if someone else do it. D.Lazard (talk) 11:14, 6 October 2023 (UTC)

I believe that it would be better not to define in terms of a specific basis. Perhaps have a section called Informal definition at the beginning of Motivation and say that it's an algebraic abstraction from oriented areas. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:46, 6 October 2023 (UTC).

I also don't think you should invoke a basis as a fundamental definition, if possible. –jacobolus (t) 14:57, 6 October 2023 (UTC)

I agree that, conceptually, a coordinate free definition is much better. But, as it is generally the case conceptual simplicity involves more abstractness and more technical background. A typical example is the definition of a real affine space as $\mathbb {R} ^{n}$ with the zero and the standard basis forgotten. This is because of the technicality needed to define "forget", that many Wikipedia articles talk of $\mathbb {R} ^{n}$ as "the" affine space of dimension n. Here, the conceptually simplest definition is as a solution of a universal problem. But this needs some technicality to show the existence of a solution and to establish the properties that are needed for manipulating multivectors. Moreover, this very abstract definition requires some expertise to be really understood. So, it is difficult to introduce this definition in the first section of the body.

The definition given in the article as a "formal definition" (I never understood what could be an informal definition) is not convenient either for the first section, since it involves advanced concepts of algebra such as "tensor algebra", "generated ideal" and "quotient by an ideal".

So, we are left with the definition that prevailed before Bourbaki, which it the definition that I have sketched above. Also, this definition is especially important since it is used in most reasoning and computations with the exterior algebra.

So, I suggest to keep the above definition from a basis, and to precede it by something like There are several equivalent definitions of the exterior algebra (see below). The one that is presented here depends formally on the choice of a basis of the vector space, but this choice is not meaningful, since the resulting exterior algebra is essentially invariant under a change of basis. D.Lazard (talk) 15:58, 6 October 2023 (UTC)

I'll try to think about what the most accessible basis-free version would be. Grassmann's 1862 Ausdehnungslehre starts with a definition of a "domain" (n-dimensional real vector space) in terms of a basis but almost immediately (§24) proves that any $n$ "magnitudes of first order" which "stand in no numerical relation to one another" (linearly independent) serve as a basis. This happens before the definition of the exterior product. –jacobolus (t) 16:59, 6 October 2023 (UTC)

Grassmann's 1844 Ausdehnungslehre doesn't lead with a basis (doesn't introduce the concept of a basis or coordinates until §87, halfway through part 1), but the presentation is geometrically motivated and not nearly concise enough for us here. It would be worth citing or possibly even substantially quoting from in footnote(s) though. –jacobolus (t) 17:27, 6 October 2023 (UTC)

For what it's worth, I strongly dislike Berger's way of introducing affine spaces in his book (not unique to Berger, but Wikipedia cites him). It is done for his own convenience in the context of a logical structure built around coordinates as a fundamental abstraction, and this is relatively more concise to make rigorous, but I think it does a disservice to his students and Berger doesn't spend enough effort on emphasizing that this is just one arbitrary definition and that coordinates are best thought of as a tool of convenience rather than an essential foundation. YMMV. –jacobolus (t) 17:15, 6 October 2023 (UTC)

I strongly propose including div/grad/curl, in an informal way, in the motivating section (maybe not the first part), c.f. my first talk post about editing the introduction. MeowMathematics (talk) 12:14, 7 October 2023 (UTC)

You might be looking for geometric calculus. These are all dependent on a metrical structure. (Curl not essentially so, if you use the wedge product rather than cross product). –jacobolus (t) 05:24, 9 October 2023 (UTC)

I agree, and I would be tempted to put curl and grad as the first examples in motivations. Divergence relies on a metric, so I would put it a bit later. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:33, 11 October 2023 (UTC)

I've made a ham-fisted first attempt at the reorganisation, and rewriting the definition section. Others feel free to improve. In particular, someone please write a basis-dependent definition in front of the current definition. MeowMathematics (talk) 22:17, 8 October 2023 (UTC)

I D.Lazard's take on the lead is a clearer basic summary. YMMV. –jacobolus (t) 05:27, 9 October 2023 (UTC)

This is not the talk thread about the lead, but in any case feel free to make concrete suggestions. MeowMathematics (talk) 11:24, 9 October 2023 (UTC)

I consider Recall normal in mathematical literature, but I believe that it is problematical in Wiki. Also, I suggest that you start the formal definition section with There are several equivalent definitions of the exterior product. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:33, 11 October 2023 (UTC)

Why not use the simpler quotient?

The article currently constructs the exterior algebra as the quotient of the tensor algebra by the two-sided ideal generated by elements of the form $u\otimes v+v\otimes u$. That seems unnecessarily complicated: With $u=a+b$ and $v=a-b$ we have

$u\otimes v+v\otimes u=(a+b)\otimes (a-b)+(a-b)\otimes (a+b)=2a\otimes a-2b\otimes b\;,$

so this ideal is also generated by elements of the form $v\otimes v$. Is there a reason not to use this simpler construction?

Joriki (talk) 20:03, 21 November 2023 (UTC)

Joriki, your observation is valid. There is no reason to use the more complicated construction, which is also incorrect in characteristic 2. This is the result of a recent substantial revision. —Quondum 18:26, 10 December 2023 (UTC)

 Done —Quondum 20:52, 11 December 2023 (UTC)

Exterior products

In elementary differential geometry and mathematical physics texts the exterior product seemed to more often be defined as the "determinant convention":

$\omega \wedge \eta ={\frac {(k+m)!}{k!\,m!}}\operatorname {Alt} (\omega \otimes \eta ),$

as opposed to the "Alt convention":

$\omega \wedge \eta =\operatorname {Alt} (\omega \otimes \eta )$

There is a discussion of these two "conventions" in John M. Lee, "Introduction to Smooth Manifolds" (2nd Edition) p.358.

The choice of which definition to use is largely a matter of taste. Although the definition of the Alt convention is perhaps a bit more natural, the computational advantages of the determinant convention make it preferable for most applications, and we use it exclusively in this book. (But see Problem 14-3 for an argument in favor of the Alt convention.) The determinant convention is most common in introductory differential geometry texts, and is used, for example, in [Boo86, Cha06, dC92, LeeJeff09, Pet06, Spi99]. The Alt convention is used in [KN69] and is more common in complex differential geometry.

While the current article does have a section Alternating multilinear forms it is deep in the article and its practical relevance is not pointed out. I suspect a majority of people consulting Wikipedia about exterior products are taking courses or reviewing material in elementary differential geometry or physics and would appreciate an additional section, located near the beginning of the article, along the lines of Lee's textbook or other introductory presentations. Pmokeefe (talk) 15:26, 28 December 2023 (UTC)

This is a fair enough observation, though it is based on an essentially a distinct definition (making it more than just a convention), where we delineate the alternating subset of the tensor algebra over a field of characteristic zero, on which define a new operation $\wedge$. This is not directly equivalent to the approach using the quotient by the ideal, and requires characteristic 0. So this veers into pedagogy, where we would be addressing the embedding of the exterior algebra into the tensor algebra. Given the nonequivalence and the need to explain the inherent difference, it seems difficult to do more than put a mention of the section near the beginning. —Quondum 21:41, 28 December 2023 (UTC)

The distinction is more or less just a matter of whether you consider a wedge product to represent a simplex or a parallelotope / what you consider the basic unit for hypervolume to be. If you want to relate the simplex and the parallelotope with the same specific corner, you have to multiply or divide by this scaling factor.

But if we are defining the wedge product of vectors to "be" its own new kind of object (a "blade") without explicitly basing it on previously defined concepts, then it doesn't really have any inherent unit, and whether you consider it to represent a simplex or a parallelotope doesn't change the algebra in any meaningful way (if you make up a basis and start trying to do concrete computations for solving some practical problem, you may need to pick an interpretation). –jacobolus (t) 20:25, 29 December 2023 (UTC)

Summations

This is an excellent article.

But it would be improved if not just some, but all, summations were denoted by the uppercase 𝚺 notation.

Currently there are several places where the Einstein summation convention is used instead, with virtually no explanation.

It would be much better if all summations are denoted by 𝚺.

The 𝚺 summation notation is understood by all disciplines that use mathematics.

The Einstein summation convention is not.

— Preceding unsigned comment added by 2601:200:c082:2ea0:2df9:7a03:f281:1107 (talk) 17:34, 18 March 2024 (UTC)

Please use <math>\Sigma</math> for $\Sigma$; not everybody has the correct Unicode fonts to handle Sigma.

The Einstein Summation Convention is omnipresent in the relevant fields, even if there are fields where it is less common. I would suggest just adding a brief explanation. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:07, 19 March 2024 (UTC)

This article only uses the summation convention in a few places. One in the section on index notation, where it is well-justified, but should be explained and linked, and as far as I can tell only in one other place where it is not very essential. Tito Omburo (talk) 15:35, 19 March 2024 (UTC)

I added a link for index notation. What is the other section? Thanks. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:53, 19 March 2024 (UTC)

Very briefly in applications where the electromagnetic field is discussed. Tito Omburo (talk) 09:12, 22 March 2024 (UTC)

Thanks. #Electromagnetic field comes after #Index notation, which now links to the Einstein summation convention; is that good enough or should the reference be in both sections? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:14, 22 March 2024 (UTC)

On edits by Timo Omburo

It seems to me that this user rolled back all the changes that were a result of a careful discussion late last year among many editors.

The introduction to this article is, again, horrendously long. We have already discussed this. I do not want to have to rehash the same discussion every time - if they want to do extremely large edits, let them argue their case out in this talk page.

Otherwise, I am rolling back their changes in the lede, because of all the problems already listed below which they did not engage with, in a week or so.

MeowMathematics (talk) 06:54, 26 March 2024 (UTC)

Personally I think the lead in Special:PermanentLink/1178847481 (which I worked on some after D.Lazard) was a better general approach than MeowMathematics's replacement which ultimately settled at Special:PermanentLink/1195382941, or the current version Special:PermanentLink/1215275338, though I'm sure it would be possible to do better than any of these. I don't think MeowMathematics's version can really be characterized as resulting from consensus, and I intended to (someday) get around to reworking it again, but didn't have the energy to wade into. YMMV. –jacobolus (t) 09:20, 26 March 2024 (UTC)

I'm fine with that version of the lede. The MeowMathematics version is, to me, unacceptable. I've gone ahead and put that version in. Tito Omburo (talk) 12:47, 26 March 2024 (UTC)

I think this puts WP:BRD the wrong way around. There was no consensus for the "MeowMathematics" edits. The evidence of this lack of consensus is my rollback to earlier versions of things.

However, on substance, the "MeowMathematics" version was clearly inferior to what had been there before. Firstly, the lede should provide an accessible overview of the article, which the MeowMathematics version did not. Secondly, the purpose of a motivation section is to motivate the definition, so it makes sense to have it be first, before a formal definition. Thirdly, the definition in terms of formal symbols was not technically correct, and also lacked a reference. Various other issues with this article were as follows. Sources had been removed from various places, which I restored. Plucker embeddings and differential forms are discussed much later in the article, and they are out of scope for a section on motivating examples. A lot of the linear algebra section was referenced to a self-published work, and seemed out of scope for this article.

I do not see any consensus on the discussion page for MeowMathematics's edits. In fact, mostly people seemed to be at best neutral to these edits (advising them, for instance, to work on a draft before working here, or else to be bold). My changes to the lede were reverting it to an earlier consensus version based on years of discussions. My changes to the article itself mostly restored the old consensus ordering of the sections, and various consolidations of content to other parts (e.g., differential forms and Plucker embeddings to much later).

Incidentally, my version of the article is about 10% shorter than your version, and your primary complaint seems to be that the article was too long. If you wish to change back to your version from the prior consensus version, please discuss why you think yours is better. Tito Omburo (talk) 12:31, 26 March 2024 (UTC)