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. Therefore, the symmetric algebra over V can be viewed as a "coordinate free" polynomial ring over V. The symmetric algebra S(V) can be built as the quotient of the tensor algebra T(V) by the two-sided ideal generated by the elements of the form x ⊗ y − y ⊗ x. All these definitions and properties extend naturally to the case where V is a module (not necessarily a free one) over a commutative ring.
Construction
From tensor algebra It is possible to use the tensor algebra T(V) to describe the symmetric algebra S(V). In fact, S(V) can be defined as the quotient algebra of T(V) by the two-sided ideal generated by the commutators v ⊗ w − w ⊗ v . {\displaystyle v\otimes w-w\otimes v.}
It is straightforward to verify that the resulting algebra satisfies the universal property stated in the introduction. Because of the universal property of the tensor algebra, a linear map f from V to a commutative algebra A extends to an algebra homomorphism T ( V ) → A {\displaystyle T(V)\rightarrow A} , which factors through S(V) because A is commutative. The extension of f to an algebra homomorphism S ( V ) → A {\displaystyle S(V)\rightarrow A} is unique because V generates S(V) as a K-algebra. 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. Here, the forgetful functor from commutative algebras to vector spaces or modules (forgetting the multiplication) is the composition of the forgetful functors from commutative algebras to associative algebras (forgetting commutativity), and from associative algebras to vectors or modules (forgetting the multiplication). 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.
From polynomial ring 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 implies that K[B] and S(V) are canonically isomorphic, and can therefore be identified. This results also immediately from general considerations of category theory, since free modules and polynomial rings are free objects of their respective categories. If V is a module that is not free, it can be written V = L / M , {\displaystyle V=L/M,} where L is a free module, and M is a submodule of L. In this case, one has
S ( V ) = S ( L / M ) = S ( L ) / ⟨ M ⟩ , {\displaystyle S(V)=S(L/M)=S(L)/\langle M\rangle ,}
where ⟨ M ⟩ {\displaystyle \langle M\rangle } is the ideal generated by M. (Here, equals signs mean equality up to a canonical isomorphism.) 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.)
Grading The symmetric algebra is a graded algebra. That is, it is a direct sum
S ( V ) = ⨁ n = 0 ∞ S n ( V ) , {\displaystyle S(V)=\bigoplus _{n=0}^{\infty }S^{n}(V),}
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