-monoid

Tensor algebra

The tensor algebra of a vector space is the direct sum of all tensor powers of together with the outer product , #m/def/ralg i.e. denoting ,

where . The tensor algebra is a very simple -monoid1 and -graded algebra.

Universal property

The tensor algebra has a unique extension to a functor so that the canonical inclusion becomes a natural transformation , where is the forgetful functor (thus creating a Free-forgetful adjunction). This is enabled by characterising with the following universal property:

If and is a linear map of vector spaces there exists a unique so that , i.e. the following diagram commutes

https://q.uiver.app/#q=WzAsNSxbMiwwLCJGVF5cXGJ1bGxldCBWIl0sWzIsMiwiRkEiXSxbMCwwLCJWIl0sWzQsMCwiVF5cXGJ1bGxldCBWIl0sWzQsMiwiQSJdLFsyLDAsIlxcaW90YV9WIl0sWzIsMSwiZiIsMl0sWzAsMSwiRiBcXGJhciBmIl0sWzMsNCwiXFxiYXIgZiJdXQ==

Proof

#missing/proof

Graded structure

The tensor algebra is -graded, since . If is itself a -graded vector space for some monoid , then possesses an additional unique gradation extending that of so that .

#state/tidy | #lang/en | #SemBr

Footnotes

  1. Indeed, there is a sense in which it is the most simple, i.e. a Free-forgetful adjunction. ↩