K-monoid

Symmetric algebra

The symmetric algebra of a vector space is the universal commutative K-monoid containing , as formalized by the Universal property. Compare this to the exterior algebra, which is has the alternating property.

The symmetric algebra is in a sense generalized by, or rather quantized by, the Weyl algebra. Conceptually similar is the Exterior algebra.

Universal property

The symmetric algebra is a pair consisting of a commutative K-monoid and a linear map such that given any commutative unital associative algebra and any linear map , there exists a unique unital algebra homomorphism for which the following diagram commutes: falg

has a unique extension to a functor such that becomes a natural transformation.

Construction

The symmetric algebra may be constructed as a quotient of the tensor algebra

where the divisor is the algebra ideal generated by tensors of the form , where the symmetric product is the quotient algebra product.

Proof of universal property

proof

Graded structure

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

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