In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and is, in some sense, minimal for this property.
Here, "minimal" means that S(V) satisfies the following universal property: for every linear map f from V to a commutative algebra A, there is a unique algebra homomorphism g : S(V) → A such that f = g ∘ i, where i is the inclusion map of V in S(V).
If B is a basis of V, the symmetric algebra S(V) can be identified, through a canonical isomorphism, to the polynomial ring K[B], where the elements of B are considered as indeterminates.
All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring.
In fact, S(V) can be defined as the quotient algebra of T(V) by the two-sided ideal generated by the commutators
It is straightforward to verify that the resulting algebra satisfies the universal property stated in the introduction.
This results also directly from a general result of category theory, which asserts that the composition of two left adjoint functors is also a left adjoint functor.
As the tensor algebra and the quotient by commutators are left adjoint to these forgetful functors, their composition is left adjoint to the forgetful functor from commutative algebra to vectors or modules, and this proves the desired universal property.
The symmetric algebra S(V) can also be built from polynomial rings.
If V is a K-vector space or a free K-module, with a basis B, let K[B] be the polynomial ring that has the elements of B as indeterminates.
The homogeneous polynomials of degree one form a vector space or a free module that can be identified with V. It is straightforward to verify that this makes K[B] a solution to the universal problem stated in the introduction.
This results also immediately from general considerations of category theory, since free modules and polynomial rings are free objects of their respective categories.
Again this can be proved by showing that one has a solution of the universal property, and this can be done either by a straightforward but boring computation, or by using category theory, and more specifically, the fact that a quotient is the solution of the universal problem for morphisms that map to zero a given subset.
(Depending on the case, the kernel is a normal subgroup, a submodule or an ideal, and the usual definition of quotients can be viewed as a proof of the existence of a solution of the universal problem.)
called the nth symmetric power of V, is the vector subspace or submodule generated by the products of n elements of V. (The second symmetric power
One follows from the tensor-algebra construction: since the tensor algebra is graded, and the symmetric algebra is its quotient by a homogeneous ideal: the ideal generated by all
A non-free module can be written as L / M, where L is a free module of base B; its symmetric algebra is the quotient of the (graded) symmetric algebra of L (a polynomial ring) by the homogeneous ideal generated by the elements of M, which are homogeneous of degree one.
as the solution of the universal problem for n-linear symmetric functions from V into a vector space or a module, and then verify that the direct sum of all
satisfies the universal problem for the symmetric algebra.
The symmetric tensors of degree n form a vector subspace (or module) Symn(V) ⊂ Tn(V).
The symmetric tensors are the elements of the direct sum
can be non surjective; for example, over the integers, if x and y are two linearly independent elements of V = S1(V) that are not in 2V, then
Moreover, this isomorphism does not extend to the cases of fields of positive characteristic and rings that do not contain the rational numbers.
Given a module V over a commutative ring K, the symmetric algebra S(V) can be defined by the following universal property: As for every universal property, as soon as a solution exists, this defines uniquely the symmetric algebra, up to a canonical isomorphism.
This section is devoted to the main properties that belong to category theory.
The symmetric algebra is a functor from the category of K-modules to the category of K-commutative algebra, since the universal property implies that every module homomorphism
The universal property can be reformulated by saying that the symmetric algebra is a left adjoint to the forgetful functor that sends a commutative algebra to its underlying module.
One can analogously construct the symmetric algebra on an affine space.
For instance, given a linear polynomial on a vector space, one can determine its constant part by evaluating at 0.
The Sk are functors comparable to the exterior powers; here, though, the dimension grows with k; it is given by where n is the dimension of V. This binomial coefficient is the number of n-variable monomials of degree k. In fact, the symmetric algebra and the exterior algebra appear as the isotypical components of the trivial and sign representation of the action of