Algebras and modules
Model category presentations
Geometry on formal duals of algebras
With duals for objects
With duals for morphisms
Special sorts of products
In higher category theory
The symmetric algebra of a vector space is the free commutative algebra over .
This construction generalizes to group representations, chain complexes, vector bundles, coherent sheaves, and indeed objects in any symmetric monoidal linear categories with enough colimits, where the tensor product distributes over those colimits (as in a 2-rig).
We begin with the construction for vector spaces and then sketch how to generalize it.
For vector spaces
Suppose is a vector space over a field . Then the symmetric algebra is generated by the elements of using these operations:
- addition and scalar multiplication
- an associative binary operation
subject to these identities:
- the identities necessary for to be an associative algebra
- the identity for all .
It then follows that is a graded algebra where is spanned by -fold products, that is, elements of the form
where . Clearly is also commutative.
The symmetric algebra of is also denoted . It is also called the polynomial algebra. However we should carefully distinguish between polynomials in the elements of , which form the algebra , and polynomial functions on the vector space , which form the algebra . In quantum physics, a similar construction for Hilbert spaces is known as the Fock space.
More generally, suppose is any symmetric monoidal category and is any object. Then we can form the tensor powers . If has countable coproducts we can form the coproduct
(which we write here as a direct sum), and if the tensor product distributes over these coproducts, becomes a monoid object in , with multiplication given by the obvious maps
This monoid object is called the tensor algebra of .
The symmetric group acts on , and if is a linear category over a field of characteristic zero, then we can form the symmetrization map
This is an idempotent, so if idempotents split in we can form its cokernel, called the th symmetric tensor power or symmetric power . The coproduct
becomes a monoid object called the symmetric algebra of .
If is a more general sort of symmetric monoidal category, then we need a different construction of . For example, if is a symmetric monoidal category with finite colimits, we can simply define to be the coequalizer of the action of the symmetric group on . And if also has countable coproducts, we can define
Then, if the tensor product distributes over these colimits (as in a 2-rig), will become a commutative monoid object in . In fact, it will be the free commutative monoid object on , meaning that any morphism in
where is a commutative monoid, factors uniquely as the obvious morphism
followed by a morphism of commutative monoids
as in this commutative triangle: