Symmetric algebra

The symmetric algebra is a graded algebra. That is, it is a direct sum

${\displaystyle S(V)=\bigoplus _{n=0}^{\infty }S^{n}(V),}$

where ${\displaystyle S^{n}(V),}$ 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 ${\displaystyle S^{2}(V)}$ is sometimes called the symmetric square of V).

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 a homogeneous ideal: the ideal generated by all ${\displaystyle x\otimes y-y\otimes x,}$ where x and y are in V, 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 the total degree. 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.

One can also define ${\displaystyle S^{n}(V)}$ 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 ${\displaystyle S^{n}(V)}$ 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 a symmetric tensor. However, symmetric tensors are strongly related to the symmetric algebra.

A symmetric tensor of degree n is an element of Tⁿ(V) that is invariant under the action of the symmetric 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 linear endomorphism of Tⁿ(V). A symmetric tensor is a tensor that is invariant under all these endomorphisms. 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 ${\displaystyle \textstyle \bigoplus _{n=0}^{\infty }\operatorname {Sym} ^{n}(V),}$ which is a graded vector space (or a graded 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 to Symⁿ(V) of the canonical surjection ${\displaystyle T^{n}(V)\to S^{n}(V).}$ If n! is invertible in the ground field (or ring), then ${\displaystyle \pi _{n}}$ is an isomorphism. This is always the case with a ground field of characteristic zero. The inverse isomorphism is the linear map defined (on products of n vectors) by the symmetrization

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

The map ${\displaystyle \pi _{n}}$ is not injective if the characteristic is less than n+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, if x and y are two linearly independent elements of V = S¹(V) that are not in 2V, then ${\displaystyle xy\not \in \pi _{n}(\operatorname {Sym} ^{2}(V)),}$ since ${\displaystyle {\frac {1}{2}}(x\otimes y+y\otimes x)\not \in \operatorname {Sym} ^{2}(V).}$

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 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:

For every K-linear map f from V to a commutative K-algebra A, there is a unique K-algebra homomorphism ${\displaystyle g:S(V)\to A}$ such that ${\displaystyle f=g\circ i,}$ where i is the inclusion of V in S(V).

As for every universal property, as soon as a solution exists, this defines uniquely the symmetric algebra, up to a canonical 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 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 ${\displaystyle f:V\to W}$ can be uniquely extended to an algebra homomorphism ${\displaystyle S(f):S(V)\to S(W).}$

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. The key difference is that the symmetric algebra of an affine space is not a graded algebra, but a filtered 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).

The S^k are functors comparable to the exterior powers; here, though, the dimension grows with k; it is given by

${\displaystyle \operatorname {dim} (S^{k}(V))={\binom {n+k-1}{k}}}$

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 ${\displaystyle S_{n}}$ acting on the tensor product ${\displaystyle V^{\otimes n}}$ (for example over the complex field)

The symmetric algebra can be given the structure of a Hopf algebra. See Tensor algebra for details.

The symmetric algebra S(V) is the universal enveloping algebra of an abelian Lie algebra, i.e. one in which the Lie bracket is identically 0.

• exterior algebra, the alternating algebra analog
• graded-symmetric algebra, a common generalization of a symmetric algebra and an exterior algebra
• Weyl algebra, a quantum deformation of the symmetric algebra by a symplectic form
• Clifford algebra, a quantum deformation of the exterior algebra by a quadratic form

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