Symmetric algebra

We pass from the tensor algebra to the symmetric algebra by forcing it to be commutative.

In mathematics, the **symmetric algebra** *S*(*V*) (also denoted Sym(*V*)) on a vector space *V* over a field *K* is the free commutative unital associative algebra over *K* containing *V*.

It corresponds to polynomials with indeterminates in *V*, without choosing coordinates. The dual, *S*(*V*∗) corresponds to polynomials *on* *V*.

A Frobenius algebra whose bilinear form is symmetric is also called a *symmetric algebra*, but is not discussed here.

## Construction

It is possible to use the tensor algebra *T*(*V*) to describe the symmetric algebra *S*(*V*). In fact we pass from the tensor algebra to the symmetric algebra by forcing it to be commutative; if elements of *V* commute, then tensors in them must, so that we construct the symmetric algebra from the tensor algebra by taking the quotient algebra of *T*(*V*) by the ideal generated by the differences of products (i.e. the  Commutator)

v
⊗
w
−
w
⊗
v
.

{\displaystyle v\otimes w-w\otimes v.}

for all *v* and *w* in *V*.

In other words, if the commutator were already zero (as it would be if *T* were the set of all diagonal matrices) then that would indicate that *T* is already fully commutative (symmetric) and we would be done. When the commutator isnt zero then we just define it to be zero. (Just as

0
≡
2

(
mod

2
)

{\displaystyle 0\equiv 2{\pmod {2}}}

) and then reduce *T* to the quotient group.

In effect, *S*(*V*) is the same as the polynomial ring over *K* in indeterminates that are a basis for *V*.

### Grading

Just as with a polynomial ring, there is a direct sum decomposition of *S*(*V*) as a graded algebra, into summands

*Sk*(*V*)
which consist of the linear span of the monomials in vectors of *V* of degree *k*, for *k* = 0, 1, 2, ... (with *S*0(*V*) = *K* and *S*1(*V*) = *V*). The *K*-vector space *Sk*(*V*) is the ***k*-th symmetric power** of *V*. (The case *k* = 2, for example, is the **symmetric square** and denoted Sym2(*V*).) It has a universal property with respect to symmetric multilinear operators defined on *V**k*.

In terms of the tensor algebra grading, *Sk*(*V*) is the quotient space of *Tk*(*V*) by the subspace generated by all differences of products

v
⊗
w
−
w
⊗
v
.

{\displaystyle v\otimes w-w\otimes v.}

and products of these with other algebra elements.

### Distinction from symmetric tensors

The symmetric algebra and symmetric tensors are easily confused: the symmetric algebra is a *quotient* of the tensor algebra, while the symmetric tensors are a *subspace* of the tensor algebra.

By expressing the symmetric algebra as a quotient, it inherits the universal property already present in the tensor algebra; that is, quotienting preserves the universal property. That is, the symmetric algebra is the "most general" symmetric algebra, in that any other symmetric algebra (defined by some symmetric product

F
(
V
,
W
)

{\displaystyle F(V,W)}

) is homomorphic to the symmetric algebra.

Symmetric tensors are defined as invariants: given the natural action of the symmetric group on the tensor algebra, the symmetric tensors are the subspace on which the symmetric group acts trivially. Note that under the tensor product, symmetric tensors are not a subalgebra: given linearly independent vectors *v* and *w*, they are trivially symmetric 1-tensors, but *v* ⊗ *w* is not a symmetric 2-tensor.

The grade 2 part of this distinction is the difference between symmetric bilinear forms (symmetric 2-tensors) and quadratic forms (elements of the symmetric square), as described in ε-quadratic forms.

In characteristic 0, symmetric tensors and the symmetric algebra can be identified. In any characteristic, there is a symmetrization map from the symmetric algebra to the symmetric tensors, given by:

v

1

⋯

v

k

↦

∑

σ
∈

S

k

v

σ
(
1
)

⊗
⋯
⊗

v

σ
(
k
)

.

{\displaystyle v\_{1}\cdots v\_{k}\mapsto \sum \_{\sigma \in S\_{k}}v\_{\sigma (1)}\otimes \cdots \otimes v\_{\sigma (k)}.}

The composition of this map with the inclusion of the symmetric tensors in the tensor algebra and the quotient to the symmetric algebra is multiplication by *k*! on the *k*th graded component of the symmetric tensors.

Thus in characteristic 0, the symmetrization map is an isomorphism of graded vector spaces, and one can identify symmetric tensors with elements of the symmetric algebra. One divides by *k*! to make this a section of the quotient map:

v

1

⋯

v

k

↦

1

k
!

∑

σ
∈

S

k

v

σ
(
1
)

⊗
⋯
⊗

v

σ
(
k
)

.

{\displaystyle v\_{1}\cdots v\_{k}\mapsto {\frac {1}{k!}}\sum \_{\sigma \in S\_{k}}v\_{\sigma (1)}\otimes \cdots \otimes v\_{\sigma (k)}.}

For instance,

v
w
↦

1
2

(
v
⊗
w
w
⊗
v
)

{\displaystyle vw\mapsto {\frac {1}{2}}(v\otimes w+w\otimes v)}

.

This is related to the representation theory of the symmetric group: in characteristic 0, over an algebraically closed field, the group algebra is semisimple, so every representation splits into a direct sum of irreducible representations, and if *T* = *S* ⊕ *V*, one can identify *S* as either a subspace of *T* or as the quotient *T*/*V*.

## Interpretation as polynomials

Main articles: Ring of polynomial functions
Given a vector space *V*, the polynomials on this space are *S*(*V*∗), the symmetric algebra of the *dual* space: a polynomial on a space *evaluates* vectors on the space, via the pairing

S
(

V

∗

)
×
V
→
K

{\displaystyle S(V^{\*})\times V\to K}

.

For instance, given the plane with a basis {(1,0), (0,1)}, the (homogeneous) linear polynomials on *K*2 are generated by the coordinate functionals *x* and *y*. These coordinates are covectors: given a vector, they evaluate to their coordinate, for instance:

x
(
2
,
3
# )
2
,

 and

y
(
2
,
3
# )

{\displaystyle x(2,3)=2,{\text{ and }}y(2,3)=3.}

Given monomials of higher degree, these are elements of various symmetric powers, and a general polynomial is an element of the symmetric algebra. Without a choice of basis for the vector space, the same holds, but one has a polynomial algebra without choice of basis.

Conversely, the symmetric algebra of the vector space itself can be interpreted, not as polynomials *on* the vector space (since one cannot evaluate an element of the symmetric algebra of a vector space against a vector in that space: there is no pairing between *S*(*V*) and *V*), but polynomials *in* the vectors, such as *v*2 − *vw* + *uv*.

### Symmetric algebra of an affine space

One can analogously construct the symmetric algebra on an affine space (or its dual, which corresponds to polynomials on that 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).

## Universal properties

The symmetric algebra on a vector space is a free object in the category of commutative unital associative algebras (in the sequel, "commutative algebras").

Formally, the map that sends a vector space to its symmetric algebra is a functor from vector spaces over *K* to commutative algebras over *K*, and is a *free object*, meaning that it is left adjoint to the forgetful functor that sends a commutative algebra to its underlying vector space.

The unit of the adjunction is the map *V* → *S*(*V*) that embeds a vector space in its symmetric algebra.

Commutative algebras are a reflective subcategory of algebras; given an algebra *A*, one can quotient out by its commutator ideal generated by *ab* – *ba*, obtaining a commutative algebra, analogously to abelianization of a group. The construction of the symmetric algebra as a quotient of the tensor algebra is, as functors, a composition of the free algebra functor with this reflection. The universal property can thus be seen as being inherited from the tensor algebra.

## Analogy with exterior algebra

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

dim
⁡
(

S

k

(
V
)
# )

(

n
k
−
1

k

)

{\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

S

n

{\displaystyle S\_{n}}

 acting on the tensor product

V

⊗
n

{\displaystyle V^{\otimes n}}

 (for example over the complex field)

## Module analog

The construction of the symmetric algebra generalizes to the symmetric algebra *S*(*M*) of a module *M* over a commutative ring. If *M* is a free module over the ring *R*, then its symmetric algebra is isomorphic to the polynomial algebra over *R* whose indeterminates are a basis of *M*, just like the symmetric algebra of a vector space. However, if *M* is not free then *S*(*M*) is more complicated.

## As a Hopf algebra

The symmetric algebra can be given the structure of a Hopf algebra. The article on the tensor algebra provides highly detailed mechanics showing how this is done.

## As a universal enveloping algebra

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.