nLab symmetric algebra

Idea

The symmetric algebra $S V$ 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 $S V$ 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 $S V$ to be an associative algebra
• the identity $v \cdot w = w \cdot v$ for all $v \in V$.

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

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

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

The symmetric algebra of $V$ is also denoted $Sym V$. It is also called the polynomial algebra. However we should carefully distinguish between polynomials in the elements of $V$, which form the algebra $S V$, 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

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

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

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

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

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

given by

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

This is an idempotent, so if idempotents split in $C$ we can form its image, called the $n$th symmetric tensor power or symmetric power $S^n V$ (e.g Gallier 2011, p. 27). The coproduct

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

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

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

Then, if the tensor product distributes over these colimits (as in a 2-rig), $S V$ 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 $C$

$$V \to A \, ,$$

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

$$V \to S V$$

followed by a morphism of commutative monoids

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

as in this commutative triangle:

$$\array { & & S V \\ & \nearrow & & \searrow \\ V & & \longrightarrow & & A }$$

In even more generality

If $V$ is an object in a CMon-enriched symmetric monoidal category $\mathcal{C}$ such that the following diagram admits a joint coequalizer for every $n \ge 2$:

(there is one morphism for every $\sigma \in \mathfrak{S}_{n}$, such permutation natural transformations being defined in the entry symmetric monoidal category),

we say that $V$ possess all the symmetric powers. The $n^{th}$ symmetric power is given by the coequalizer:

We put $S_{0}(V)=I$ and $S_{1}(V)=V$.

If the category possess the countable coproducts, then we can form

$$S(V) = \underset{n \ge 0}{\coprod}S_{n}(V)$$

As above, if the tensor product distributes over the countable coproducts and preserve the finite colimits in each argument, then $S(V)$ will be the free commutative monoid object on V.

For example, in the category Rel of sets and relations, this construction gives the set $\mathcal{M}(V)$ of multisets with elements in the set $V$.

Examples

Example

(symmetric algebra in chain complexes is differential graded-commutative algebra)

Let the ambient category be the category of cochain complexes over a ground field of characteristic zero, regarded as a symmetric monoidal category via the tensor product of chain complexes.

Then for $(V^\bullet, d)$ a cochain complex, the symmetric algebra

$$Sym\left( (V^\bullet, d) \right) \;\in\; CMon( Ch^\bullet, \otimes )$$

is the differential graded-commutative algebra whose underlying graded algebra is the graded-commutative algebra on $V^\bullet$, and whose differential is the original $d$, extended, uniquely, as a graded derivation of degree +1.

Properties

Proposition

Let the ambient category be the category of cochain complexes over a ground field of characteristic zero, regarded as a symmetric monoidal category via the tensor product of chain complexes and consider the differential graded-commutative algebra $Sym(V^\bullet,d)$ free on a cochain complex $(V^\bullet,d)$ from example .

Then the cochain cohomology (of the underlying cochain complex) of $Sym(V^\bullet,d)$ is the graded symmetric algebra on the cochain cohomology of $(V^\bullet,d)$:

$$H^\bullet\left( Sym\left(V^\bullet,d\right) \right) \;\simeq\; Sym\left( H^\bullet(V^\bullet,d) \right) \,.$$

See also this MO discussion.

Proof

We have the following sequence of linear isomorphisms:

$$\begin{aligned} H^\bullet \left( Sym( V^\bullet,d ) \right) & \simeq H^\bullet \left( \underset{k \in \mathbb{N}}{\oplus} \left( (V^\bullet,d)^{\otimes_k} \right)^{\Sigma_k} \right) \\ & \simeq \underset{k \in \mathbb{N}}{\oplus} H^\bullet\left( \left( (V^\bullet,d)^{\otimes_k}\right)^{\Sigma_k} \right) \\ & \simeq \underset{k \in \mathbb{N}}{\oplus} H^\bullet\left( \left( (V^\bullet,d)^{\otimes_k}\right) \right)^{\Sigma_k} \\ & \simeq \underset{k \in \mathbb{N}}{\oplus} \left( H^\bullet\left( V^\bullet,d \right)^{\otimes_k} \right)^{\Sigma_k} \\ & = Sym\left( H^\bullet(V^\bullet,d)\right) \end{aligned}$$

Here:

1. the first step uses that, while a priori the symmetric algebra is equivalently the quotient of the tensor algebra by the symmetric group action, in characteristic zero this is equivalently invariants of the symmetric group action, because here $V^G \to V \to V_G$ is a linear isomorphism;

2. the second step uses that cochain cohomology respects direct sums;

3. the third step uses that for finite groups in characteristic zero, taking invariants is compatible with passing to cochain cohomology (this prop.):

4. the fourth step is the Künneth theorem for ordinary homology over a field (this prop.).

Symmetric powers in a symmetric monoidal $\mathbb{Q}^{+}$-linear category are characterized among the countable families of objects as forming a special connected graded quasi-bialgebra (reference to come).

Related concepts

• symmetric algebra spectrum

• symmetric ring

• symmetric power

References

• Bourbaki, Algèbre, chap. III, § 6.

• Stacks Project, Tag 00DM.

• Jean Gallier, Tensor Algebras, Symmetric Algebras and Exterior Algebras [pdf], section 22 in: Notes on Differential Geometry and Lie Groups (2011)

• Jean Gallier, Jocelyn Quaintance, Tensor Algebras and Symmetric Algebras, Ch 2 in: Differential Geometry and Lie Groups – A second course, Geometry and Computing 13, Springer (2020) [doi:10.1007/978-3-030-46047-1, webpage, pdf]

On the symmetric algebra spectrum of the sphere spectrum, and its structure as a Hopf ring spectrum:

• Neil Strickland, Paul Turner, Rational Morava $E$-theory and $D S^0$, Topology 36 1 (1997) 137-151 (pdf)

Further on symmetric powers in homotopical/higher algebra:

• Sergey Gorchinskiy, Vladimir Guletskii, Symmetric powers in abstract homotopy categories, Adv. Math., 292 (2016) 707-754 [arXiv:0907.0730, doi:10.1016/j.aim.2016.01.011]

Discussion via string diagrams:

• Jean-Baptiste Vienney, String diagrams for symmetric powers I: In symmetric monoidal $\mathbb{Q}_{\geq 0}$-linear categories [arXiv:2308.02094]