Exterior Algebra

By | 2017-12-28

Interior Antiderivations

If V is a vector space then any form \omega\in V^* extends to a unique anti-derivation \imath_\omega of the exterior algebra \bigwedge(V), i.e.,

\imath_\omega(ef)=\imath_\omega(e)f+(-1)^{deg(e)}e\imath_\omega(f) for homogenous e\in\bigwedge(V)

If \{e_i\} is a basis of V, \{e^*_i\} the dual basis and if \imath_i denotes \imath_{e^*_i} we have the following action on monomials
\imath_i(e_i e_j e_k\ldots) = e_j e_k\ldots
\imath_j(e_i e_j e_k\ldots) = -e_i e_k\ldots
(The sign is negative if the number of terms preceding the matching one is odd)

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