Talk:Exterior algebra

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I might want to revive the "wedge product" page, with an eye to producing something more concrete. Would that be a problem? Gene Ward Smith 01:26, 30 Jan 2004 (UTC)

The way things are now, the main 'titles' are exterior power, tensor product and tensor algebra, and symmetric power, for one good reason. In defining the various products, from a mathematical point of view, one should explain the space in which the product has its range. This is how multilinear algebra is done, in current mathematics. It is indeed not how it is done in applications; there you define some operation, and if it is 'external' there is no immediate answer about where the product actually lies. The wedge of two vectors is just an 'expression'.

Obviously the ideal treatment gives the heuristic basis a good space. also. But as far as I'm concerned splitting up the discussion over many pages is a way of ducking the issue.

Charles Matthews 08:25, 30 Jan 2004 (UTC)

It's not clear to me if this is a yes, a no or a maybe. The point is, should there not be a place where the wedge product is defined in a way which makes sense to nonmathematicians, with an eye towards computation and applications? If so, where would this be? Gene Ward Smith 08:36, 30 Jan 2004 (UTC)

For the record, I more or less agree with Gene Ward Smith. The exterior algebra is fundamental enough to warrant some examples of the wedge product early in the article. Non-mathematicians use differential forms as well, but they aren't necessarily comfortable with terms like "unital" or "associative algebra." Wikimathematicians: please try to say what it is before delving into details. We are all guilty of this to some extent, but I've decided to tag the article. Thanks, 151.204.6.171 00:28, 5 November 2005 (UTC)

The minor (matrix) page would be a good location for that, considering that in algorithmic terms what you do is form a vector of minors.

As was found on the tensor page, there is really too much 'hanging off' multilinear algebra concepts in various directions for there to be any agreement on the correct encyclopedic approach. In this case my own inclination is to centralise material on this page.

Charles Matthews 15:39, 30 Jan 2004 (UTC)

I think we should rename this page exterior algebra (currently a redirect) instead of exterior power. It seems to make more sense to define the space in which the product lives before defining the product (See Charles' comment above). At any rate, this would be more consistent with the symmetric algebra and symmetric power pages. Normally I would just do this, but as there has been some discussion regarding the name, I thought I would ask first.

The pages currently redirecting here are:

-- Fropuff 00:38, 2004 May 22 (UTC)

One day ... we'll have all this linear algebra stuff sorted out. Yes, exterior algebra is the consistent top-level name. Charles Matthews 08:38, 22 May 2004 (UTC)

Ok, I moved the article over to exterior algebra, and rewrote most of it, starting with the algebra itself but emphasizing the rules governing the wedge product. I also redirected outer product to here. I suppose a section about applications and example calculations would be a good addition. What other applications besides differential n-forms are there? AxelBoldt 18:48, 4 Oct 2004 (UTC)

Plücker coordinates. I see my arguments above; but my reaction to having the algebra introduced, before the graded pieces, isn't so positive. Oh well, perhaps there is no ideal treatment. Charles Matthews 18:56, 4 Oct 2004 (UTC)

Contents

[edit] logical conjuction

Is this related to Logical conjunction? They use the same symbol

Not so you'd notice. Charles Matthews 20:39, 16 Nov 2004 (UTC)

I've seen the "wedge" notation \wedge^k V, is it the same as the lambda thing ΛkV?. Also, wedge product is supposed to cancel out when k > n (n being the dimension or V), so we wouldn't need to go up to infinity in the direct sum, just up to n, right?.--Xavier 08:32, 2005 Mar 31 (UTC)

Yes, both notations are used. And yes, if V happens to be finite dimensional then there are only finitely many terms in the direct sum. -- Fropuff 15:03, 2005 Mar 31 (UTC)

[edit] Algebraic properties

Overall, this looks like a nice article. But I would like to see a special section devoted to the algebraic properties of the exterior algebra. Notably missing is the interior product of the dual space of V and the exterior algebra over V. Apart from the wedge product (exterior multiplication), this is the second most important algebraic feature of the exterior algebra (consider, for instance, the Lie derivative and its formulation using the deRham operator and the interior product).

Furthermore, I'm really not eager to see this topic exported to the bialgebra stub. Rather it should be listed as an example once bialgebra has matured.151.204.6.171

I've added a section on the interior product, giving three different definitions of it. Please do whatever needs to be done with this material in order to integrate it better into the intended style of the article. 151.204.6.171

[edit] Defining properties

Previously this article stated that "alternating" meant that v\wedge v = 0, and that this entailed the other two properties (u\wedge v = - v\wedge u, v_1\wedge v_2\wedge\cdots \wedge v_k = 0 for linearly dependent vk). This is clearly false, as the zero operator (u\bullet v = 0) is associative, bilinear, and ... er, self-zeroing, I suppose. I've assumed that "alternating" actually means "anticommutative", and have edited accordingly (since the other two properties are easily derivable from this).  –Aponar Kestrel (talk) 05:38, 12 May 2006 (UTC)

Alternating does in fact mean v\wedge v = 0 whereas antisymmetric means u\wedge v = - v\wedge u. Over most fields these are equivalent since one implies the other. But over fields of characteristic 2, alternating is a stronger property than antisymmetry (every alternating form is antisymmetric but not vice-versa). This is why the definition is the way it is. See discussion at bilinear form. -- Fropuff 20:19, 12 May 2006 (UTC)

[edit] Wedge product contradiction

Consider a ^ b ^ c. This = a ^ (b ^ c) = a ^ (-c ^ b) = -(a ^ c) ^ b = -(-c ^ a) ^ b = c ^ a ^ b. This also = (a ^ b) ^ c = -c ^ (a ^ b) = -c ^ a ^ b. Therefore c ^ a ^ b = -c ^ a ^ b and so any triple wedge product is zero. But I do not think that this is true! Please point out the error. AeroSpace 15:59, 29 May 2006 (UTC)

I think the wedge product of two vectors is not itself a vector, so the step (a ^ b) ^ c = -c ^ (a ^ b) is incorrect. In fact, (a ^ b) ^ c = c ^ (a ^ b). Someone please correct me if I'm wrong. —Keenan Pepper 19:47, 29 May 2006 (UTC)
Yes that is right, if a is a p form and b is a q form then a ^ b = ((-1)^pq) b ^ a. AeroSpace 06:36, 30 May 2006 (UTC)

[edit] Universal property and construction

Under this heading, we find, in passing: ... We thus take the two-sided ideal I in T(V) generated by all elements of the form v⊗v for v in V, ...

Now the character I see for the product in V appears as an "unknown character" empty square when I view the page in UTF-8. Could we please replace this by whatever character is customary for Wikimathematicians to use for the product (presumably the dot product) in a vector space? yoyo 14:16, 21 August 2006 (UTC)

[edit] Example: the exterior algebra of Euclidean 3-space

This example seems to jump from the abstract definition of a wedge product to an algorithm for computing it given two vectors. Could someone fill in that leap? —Ben FrantzDale 20:35, 11 September 2006 (UTC)

not quite sure what you mean there, what's the leap? the wedge product distributes over addition so one simply multiplies according to the definition. Mct mht 07:16, 13 September 2006 (UTC)

[edit] Parallelogram

The wedge product of two vectors does not represent a parallelogram, but just the whole plane containing them, and the area of that parallelogram. That's because the wedge product of any pair of independent vectors (yielding the same area) is the same algebraic object. So what is said at the beginning is misleading. Ylebru 16:28, 31 December 2006 (UTC)

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