nLab
symmetric algebra

Idea

The symmetric algebra $SV$ of a vector space is the free commutative algebra over $V$.

This construction generalizes to group representations, chain complexes, vector bundles, coherent sheaves, and indeed objects in any symmetric monoidal linear categories with enough colimits, where the tensor product distributes over those colimits (as in a 2-rig).

Explicit definition

We begin with the construction for vector spaces and then sketch how to generalize it.

For vector spaces

Suppose $V$ is a vector space over a field $K$. Then the symmetric algebra $SV$ is generated by the elements of $V$ using these operations:

• addition and scalar multiplication
• an associative binary operation $\cdot$

subject to these identities:

• the identities necessary for $SV$ to be an associative algebra
• the identity $v\cdot w=w\cdot v$ for all $v\in V$.

It then follows that $SV$ is a graded algebra where $S^{p}V$ is spanned by $p$-fold products, that is, elements of the form

$$v_1\cdot \cdots \cdot v_p$$v_1 \cdot \cdots \cdot v_p

where $v_1,\dots ,v_p\in V$. Clearly $SV$ is also commutative.

The symmetric algebra of $V$ is also denoted $\mathrm{Sym}V$. It is also called the polynomial algebra. However we should carefully distinguish between polynomials in the elements of $V$, which form the algebra $SV$, and polynomial functions on the vector space $V$, which form the algebra $S(V^{*})$. In quantum physics, a similar construction for Hilbert spaces is known as the Fock space?.

In general

More generally, suppose $C$ is any symmetric monoidal category and $V\in C$ is any object. Then we can form the tensor powers $V^{\otimes n}$. If $C$ has countable coproducts we can form the coproduct

$$TV=\underset{n\ge 0}{⨁}{V}^{\otimes n}$$T V = \bigoplus_{n \ge 0} V^{\otimes n}

(which we write here as a direct sum), and if the tensor product distributes over these coproducts, $TV$ becomes a monoid object in $C$, with multiplication given by the obvious maps

$$V^{\otimes p}\otimes V^{\otimes q}\to V^{\otimes (p+q)}$$V^{\otimes p} \otimes V^{\otimes q} \to V^{\otimes (p+q)}

This monoid object is called the tensor algebra of $V$.

The symmetric group $S_n$ acts on $V^{\otimes n}$, and if $C$ is a linear category over a field of characteristic zero, then we can form the symmetrization map

$$p_A:V^{\otimes n}\to V^{\otimes n}$$p_A : V^{\otimes n} \to V^{\otimes n}

given by

$$p_A=\frac{1}{n!}\sum _{\sigma \in S_n}\sigma$$p_A = \frac{1}{n!} \sum_{\sigma \in S_n} \sigma

This is an idempotent, so if idempotents split in $C$ we can form its cokernel, called the $n$th symmetric tensor power or symmetric power $S^{n}V$. The coproduct

$$SV=\underset{n\ge 0}{⨁}{S}^{n}V$$S V = \bigoplus_{n \ge 0} S^n V

becomes a monoid object called the symmetric algebra of $V$.

If $C$ is a more general sort of symmetric monoidal category, then we need a different construction of $S^{n}V$. For example, if $C$ is a symmetric monoidal category with finite colimits, we can simply define $S^{n}V$ to be the coequalizer of the action of the symmetric group $S_n$ on $V^{\otimes n}$. And if $C$ also has countable coproducts, we can define

$$SV=\coprod _{n\ge 0}{S}^{n}V$$S V = \coprod_{n \ge 0} S^n V

Then, if the tensor product distributes over these colimits (as in a 2-rig), $SV$ will become a commutative monoid object in $C$. In fact, it will be the free commutative monoid object on $V$, meaning that any morphism in $C$

$$V\to A,$$V \to A \, ,

where $A$ is a commutative monoid, factors uniquely as the obvious morphism

$$V\to SV$$V \to S V

followed by a morphism of commutative monoids

$$SV\to A,$$S V \to A \, ,

as in this commutative triangle:

$$\begin{array}{ccc}& & SV\\ & \nearrow & & \searrow \\ V& & ⟶& & A\end{array}$$\array { & & S V \\ & \nearrow & & \searrow \\ V & & \longrightarrow & & A }