Categorical Properties
The symmetric algebra on a vector space is a free object in the category of commutative unital associative algebras (in the sequel, "commutative algebras").
Formally, the map that sends a vector space to its symmetric algebra is a functor from vector spaces over K to commutative algebras over K, and is a free object, meaning that it is left adjoint to the forgetful functor that sends a commutative algebra to its underlying vector space.
The unit of the adjunction is the map V → S(V) that embeds a vector space in its symmetric algebra.
Commutative algebras are a reflective subcategory of algebras; given an algebra A, one can quotient out by its commutator ideal generated by ab - ba, obtaining a commutative algebra, analogously to abelianization of a group. The construction of the symmetric algebra as a quotient of the tensor algebra is, as functors, a composition of the free algebra functor with this reflection.
Read more about this topic: Symmetric Algebra
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