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I'm reading "Symmetry in Mechanics" by S. Singer and I'm stuck on an exercise. It asks to find an antisymmetric bilinear form on [tex]R^4[/tex] that cannot be written as a wedge product of two covectors.

Here are my thoughts thus far: on [tex]R^2[/tex] its trivial to show that every antisymmetric bilinear form (ABF) is the product of two covectors. I just let F be an ABF and put in two vectors, v & w and expanded them in terms of a basis [tex]e_i[/tex]. I get something like:

[tex]F(e_1,e_2)(v_1w_2-v_2w_1)[/tex] which easily maps into [tex]F(e_1,e_2)dx\wedge dy[/tex] and you can split up the wedge anyway you'd like to, to make two covectors whose wedge gives the same result as F.

Now on [tex]R^3[/tex] are a little more complicated and I have a question about that as well. If I look at [tex]F(v,w)[/tex] where [tex]v,w\in R^3[/tex] and expand it in a basis [tex]e_i[/tex] I get what you'd expect, a cross product looking expression. Now if I also take the wedge product of two 1-forms on [tex]R^3[/tex], say [tex]\alpha,\beta[/tex] where [tex]\alpha = \alpha_idx^i[/tex] and [tex]\beta=\beta_idx^i[/tex] I get a 2-form with the coefficients you'd expect (they look like the components from a cross product).

Now the expansion of the ABF, F, using the basis [tex]e_i[/tex]gives me that cross product looking thing, but with some additional constants which I denote as:

[tex]F(e_i,e_j)=F_{ij}[/tex]. If I wish to take that ABF and associate it to a product of covectors I thought I could just equate coeffcients and solve. If I do that I get the following system:

[tex]\alpha_1\beta_2-\alpha_2\beta_1=F_{12}[/tex]

[tex]\alpha_2\beta_3-\alpha_3\beta_2=F_{23}[/tex]

[tex]\alpha_1\beta_3-\alpha_3\beta_1=F_{13}[/tex]

If i suppose that I pick my [tex]\alpha_k[/tex] first I get a matrix that's inconsistent. But of course I'm doing something stupid here because you're supposed to be able to associate ABF with wedge products of covectors in [tex]R^3[/tex].

So as you can see, I'm lost. I'd appreciate some direction.

Many thanks,

Kevin

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# Bilinear forms and wedge products

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