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,

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),

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^*,


where i denotes tensor contraction and V^* is the dual vector space to V.

See also

Decomposable Module, Exterior Algebra, Grassmannian, Plücker's Equations, Tensor Contraction, Vector Space, Wedge Product

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Decomposable." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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