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• CommentRowNumber1.
• CommentAuthorjim_stasheff
• CommentTimeSep 6th 2019
The n-lab is very clear on contraction as a derivation
For example, there is a contraction of a vector $X\in V$ and a $n$-form $\omega\in \Lambda V^*$:

$$(X,\omega)\mapsto \iota_X(\omega)$$

and $\iota_X: \omega\mapsto \iota_X(\omega)$ is a graded derivation of the exterior algebra of degree $-1$. This is also done for the tangent bundle which is a $C^\infty(M)$-module $V = T M$, then one gets the contraction of vector fields and differential forms. It can also be done in vector spaces, fibrewise.

Is it written somewhere about contraction of a $1$-form $\omega$ with an$n-vector$X\in \in \Lambda V\$ as a coderivation?
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