Why is the exterior algebra called the "exterior algebra?" What makes it "exterior?" Is it just because a module can be universally embedded into its exterior algebra, so one could view the exterior algebra as surrounding the module? Why is it not just called the "alternating algebra?"
2026-04-08 04:15:41.1775621741
Why is the exterior algebra called the "exterior algebra?" What makes it "exterior?"
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It was Grassmann that called it exterior because to have a non-null product the multiplicands must be geometrically one to the exterior of the other. For instance $$\mathbf{x}\wedge\mathbf{y}\wedge\mathbf{z}=0$$ if $\mathbf{x}$ lies in (is not exterior of) the subspace spanned by the $\mathbf{y}$ and $\mathbf{z}$. So the product is called exterior product, and consequently the algebra with this product is called exterior algebra.