Decomposable
A differential k-form omega of degree p in an exterior algebra ^ V is decomposable if there exist p one-forms alpha_i such that
| omega=alpha_1 ^ ... ^ alpha_p, |
(1)
|
where alpha ^ beta denotes a wedge product. Forms of degree 0, 1, dimV-1, and dimV are always decomposable. Hence the first instance of indecomposable forms occurs in R^4, in which case e_1 ^ e_2+e_3 ^ e_4 is indecomposable.
If a p-form omega has a form envelope of dimension p then it is decomposable. In fact, the one-forms in the (dual) basis to the envelope can be used as the alpha_i above.
Plücker's equations form a system of quadratic equations on the a_I in
| omega=suma_Ie_(i_1) ^ ... ^ e_(i_p), |
(2)
|
which is equivalent to omega being decomposable. Since a decomposable p-form corresponds to a p-dimensional subspace, these quadratic equations show that the Grassmannian is a projective algebraic variety. In particular, omega is decomposable if for every beta in ^ ^(p+1)V^*,
| i(i(beta)omega)omega=0, |
(3)
|
where i denotes tensor contraction and V^* is the dual vector space to V.
See also
Exterior Algebra, Grassmannian, Plücker's Equations, Tensor Contraction, Vector Space, Wedge ProductThis entry contributed by Todd Rowland
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Cite this as:
Rowland, Todd. "Decomposable." From MathWorld--A Wolfram Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Decomposable.html