Inmathematics, thesymmetric algebraS(V) (also denotedSym(V)) on avector spaceV over afieldK is acommutative algebra overK that containsV, and is, in some sense, minimal for this property. Here, "minimal" means thatS(V) satisfies the followinguniversal property: for everylinear mapf fromV to a commutative algebraA, there is a uniquealgebra homomorphismg :S(V) →A such thatf =g ∘i, wherei is theinclusion map ofV inS(V).

IfB is a basis ofV, the symmetric algebraS(V) can be identified, through acanonical isomorphism, to thepolynomial ringK[B], where the elements ofB are considered as indeterminates. Therefore, the symmetric algebra overV can be viewed as a "coordinate free" polynomial ring overV.

The symmetric algebraS(V) can be built as thequotient of thetensor algebraT(V) by thetwo-sided ideal generated by the elements of the formx ⊗y −y ⊗x.

All these definitions and properties extend naturally to the case whereV is amodule (not necessarily a free one) over acommutative ring.

It is possible to use thetensor algebraT(V) to describe the symmetric algebraS(V). In fact,S(V) can be defined as thequotient algebra ofT(V) by the two-sided ideal generated by thecommutators${\displaystyle v\otimes w-w\otimes v.}$

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 mapf fromV to a commutative algebraA extends to an algebra homomorphism${\displaystyle T(V)\to A}$, which factors throughS(V) becauseA is commutative. The extension offto an algebra homomorphism${\displaystyle S(V)\to A}$ is unique becauseV generatesS(V) as aK-algebra.

This results also directly from a general result ofcategory theory, which asserts that the composition of twoleft adjoint functors is also a left adjoint functor. Here, theforgetful 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.

The symmetric algebraS(V) can also be built frompolynomial rings.

IfV is aK-vector space or afreeK-module, with a basisB, letK[B] be the polynomial ring that has the elements ofB as indeterminates. Thehomogeneous polynomials of degree one form a vector space or a free module that can be identified withV. It is straightforward to verify that this makesK[B] a solution to the universal problem stated in the introduction. This implies thatK[B] andS(V) are canonically isomorphic, and can therefore be identified. This results also immediately from general considerations ofcategory theory, since free modules and polynomial rings arefree objects of their respective categories.

IfV is a module that is not free, it can be written${\displaystyle V=L/M,}$ whereL is a free module, andM is asubmodule ofL. In this case, one has

${\displaystyle S(V)=S(L/M)=S(L)/⟨M⟩,}$

where${\displaystyle ⟨M⟩}$ is the ideal generated byM. (Here, equals signs mean equalityup 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, thekernel is anormal 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.)

The symmetric algebra is agraded algebra. That is, it is adirect sum

${\displaystyle S(V)=\underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{S}^n(V),}$

where${\displaystyle S^n(V),}$ called thenthsymmetric power ofV, is the vector subspace or submodule generated by the products ofn elements ofV. (The second symmetric power${\displaystyle S^2(V)}$ is sometimes called thesymmetric square ofV).

This can be proved by various means. One follows from the tensor-algebra construction: since the tensor algebra is graded, and the symmetric algebra is its quotient by ahomogeneous ideal: the ideal generated by all${\displaystyle x\otimes y-y\otimes x,}$ wherex andy are inV, that is, homogeneous of degree one.

In the case of a vector space or a free module, the gradation is the gradation of the polynomials by thetotal degree. A non-free module can be written asL /M, whereL is a free module of baseB; its symmetric algebra is the quotient of the (graded) symmetric algebra ofL (a polynomial ring) by the homogeneous ideal generated by the elements ofM, which are homogeneous of degree one.

One can also define${\displaystyle S^n(V)}$ as the solution of the universal problem forn-linear symmetric functions fromV into a vector space or a module, and then verify that thedirect sum of all${\displaystyle S^n(V)}$ satisfies the universal problem for the symmetric algebra.

As the symmetric algebra of a vector space is a quotient of the tensor algebra, an element of the symmetric algebra is not a tensor, and, in particular, is not asymmetric tensor. However, symmetric tensors are strongly related to the symmetric algebra.

Asymmetric tensor of degreen is an element ofTⁿ(V) that is invariant under theaction of thesymmetric group${\displaystyle {\mathcal{S}}_n.}$ More precisely, given${\displaystyle \sigma \in {\mathcal{S}}_n,}$ the transformation${\displaystyle v_1\otimes \cdots \otimes v_n\mapsto v_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (n)}}$ defines a linearendomorphism ofTⁿ(V). A symmetric tensor is a tensor that is invariant under all these endomorphisms. The symmetric tensors of degreen form a vector subspace (or module)Symⁿ(V) ⊂Tⁿ(V). Thesymmetric tensors are the elements of thedirect sum${\displaystyle \underset{n=0}{\overset{\mathrm{\infty }}{⨁}}{\mathrm{Sym}}^n(V),}$ which is agraded vector space (or agraded module). It is not an algebra, as the tensor product of two symmetric tensors is not symmetric in general.

Let${\displaystyle {\pi }_n}$ be the restriction toSymⁿ(V) of the canonical surjection${\displaystyle T^n(V)\to S^n(V).}$ Ifn! is invertible in the ground field (or ring), then${\displaystyle {\pi }_n}$ is anisomorphism. This is always the case with a ground field ofcharacteristic zero. Theinverse isomorphism is the linear map defined (on products ofn vectors) by thesymmetrization

${\displaystyle v_1\cdots v_n\mapsto \frac{1}{n!}\sum _{\sigma \in S_n}{v}_{\sigma (1)}\otimes \cdots \otimes v_{\sigma (n)}.}$

The map${\displaystyle {\pi }_n}$ is not injective if the characteristic is less thann+1; for example${\displaystyle {\pi }_n(x\otimes y+y\otimes x)=2xy}$ is zero in characteristic two. Over a ring of characteristic zero,${\displaystyle {\pi }_n}$ can be non surjective; for example, over the integers, ifx andy are two linearly independent elements ofV =S¹(V) that are not in2V, then${\displaystyle xy\notin {\pi }_n({\mathrm{Sym}}^2(V)),}$ since${\displaystyle \frac{1}{2}(x\otimes y+y\otimes x)\notin {\mathrm{Sym}}^2(V).}$

In summary, over a field of characteristic zero, the symmetric tensors and the symmetric algebra form two isomorphic graded vector spaces. They can thus be identified as far as only the vector space structure is concerned, but they cannot be identified as soon as products are involved. Moreover, this isomorphism does not extend to the cases of fields of positive characteristic and rings that do not contain therational numbers.

Given amoduleV over acommutative ringK, the symmetric algebraS(V) can be defined by the followinguniversal property:

For everyK-linear mapf fromV to a commutativeK-algebraA, there is a uniqueK-algebra homomorphism${\displaystyle g:S(V)\to A}$ such that${\displaystyle f=g\circ i,}$ wherei is the inclusion ofV inS(V).

As for every universal property, as soon as a solution exists, this defines uniquely the symmetric algebra,up to acanonical isomorphism. It follows that all properties of the symmetric algebra can be deduced from the universal property. This section is devoted to the main properties that belong tocategory theory.

The symmetric algebra is afunctor from thecategory ofK-modules to the category ofK-commutative algebra, since the universal property implies that everymodule homomorphism${\displaystyle f:V\to W}$ can be uniquely extended to analgebra homomorphism${\displaystyle S(f):S(V)\to S(W).}$

The universal property can be reformulated by saying that the symmetric algebra is aleft adjoint to theforgetful functor that sends a commutative algebra to its underlying module.

One can analogously construct the symmetric algebra on anaffine space. The key difference is that the symmetric algebra of an affine space is not a graded algebra, but afiltered algebra: one can determine the degree of a polynomial on an affine space, but not its homogeneous parts.

For instance, given a linear polynomial on a vector space, one can determine its constant part by evaluating at 0. On an affine space, there is no distinguished point, so one cannot do this (choosing a point turns an affine space into a vector space).

TheS^k arefunctors comparable to theexterior powers; here, though, thedimension grows withk; it is given by

${\displaystyle \dim (S^k(V))=(\genfrac{}{}{0ex}{}{n+k-1}{k})}$

wheren is the dimension ofV. Thisbinomial coefficient is the number ofn-variable monomials of degreek.In fact, the symmetric algebra and the exterior algebra appear as the isotypical components of the trivial and sign representation of the action of${\displaystyle S_n}$ acting on the tensor product${\displaystyle V^{\otimes n}}$ (for example over the complex field)^{[citation needed]}

The symmetric algebra can be given the structure of aHopf algebra. SeeTensor algebra for details.

The symmetric algebraS(V) is theuniversal enveloping algebra of anabelian Lie algebra, i.e. one in which the Lie bracket is identically 0.

• exterior algebra, thealternating algebra analog
• graded-symmetric algebra, a common generalization of a symmetric algebra and an exterior algebra
• Weyl algebra, aquantum deformation of the symmetric algebra by asymplectic form
• Clifford algebra, aquantum deformation of the exterior algebra by aquadratic form
• Proj construction § Proj of a quasi-coherent sheaf, an application of symmetric algebras in algebraic geometry

• Bourbaki, Nicolas (1989),Elements of mathematics, Algebra I, Springer-Verlag,ISBN 3-540-64243-9

• Algebras
• Multilinear algebra
• Polynomials
• Ring theory

• Use American English from February 2019
• All Wikipedia articles written in American English
• Articles with short description
• Short description is different from Wikidata
• All articles with unsourced statements
• Articles with unsourced statements from December 2019