K-monoid

Tensor algebra

The tensor algebra of a vector space is the direct sum of all tensor powers of together with the outer product , falg i.e. denoting ,

where . The tensor algebra is a very simple K-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

Proof

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 .

Related

• Exterior algebra
• Symmetric algebra
• Universal enveloping algebra

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Footnotes

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