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ac.commutative algebra – Algebraic construction of the house of multiaffine maps

Let $V$ breathe a vector house over a bailiwick $mathbb F$ and $ok$ some unaffected quantity.

It is not difficult to display that the house of multiaffine maps $V^{[k]}tomathbb F$ decomposes as a direct sum of vector areas $bigoplus_{Isubset [k]} M_I $ the place $M_I$ is the house of multilinear maps $V^Itomathbb F$ (considered maps on $V^{[k]}$ which solely concern concerning the coordinates in $I$).

But there may be too extra construction right here – if $I,J$ are disjoint we’ve an patent bilinear map $M_Itimes M_Jto M_{Isqcup J}.$ These maps are too associative (when $I,J,Okay$ are pairwise disjoint) and commutative.

So is there a designation for this kind of algebraic construction? If so, the place may I examine it?

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