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symmetric algebra

Context

Algebra

Monoidal categories

Contents

Idea

The symmetric algebra SVS V of a vector space is the free commutative algebra over VV.

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 VV is a vector space over a field KK. Then the symmetric algebra SVS V is generated by the elements of VV using these operations:

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

subject to these identities:

  • the identities necessary for SVS V to be an associative algebra
  • the identity vw=wvv \cdot w = w \cdot v for all vVv \in V.

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

v 1v pv_1 \cdot \cdots \cdot v_p

where v 1,,v pVv_1, \dots, v_p \in V. Clearly SVS V is also commutative.

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

In general

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

TV= n0V 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, TVT V becomes a monoid object in CC, with multiplication given by the obvious maps

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

This monoid object is called the tensor algebra of VV.

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

p A:V nV n p_A : V^{\otimes n} \to V^{\otimes n}

given by

p A=1n! σS nσ p_A = \frac{1}{n!} \sum_{\sigma \in S_n} \sigma

This is an idempotent, so if idempotents split in CC we can form its cokernel, called the nnth symmetric tensor power or symmetric power S nVS^n V. The coproduct

SV= n0S nV S V = \bigoplus_{n \ge 0} S^n V

becomes a monoid object called the symmetric algebra of VV.

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

SV= n0S nV S V = \coprod_{n \ge 0} S^n V

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

VA, V \to A \, ,

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

VSV V \to S V

followed by a morphism of commutative monoids

SVA, S V \to A \, ,

as in this commutative triangle:

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

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

Sym((V ,d))CMon(Ch ,) 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 V^\bullet, and whose differential is the original dd, 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 ,d)Sym(V^\bullet,d) free on a cochain complex (V ,d)(V^\bullet,d) from example 1.

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

H (Sym(V ,d))Sym(H (V ,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:

H (Sym(V ,d)) H (k((V ,d) k) Σ k) kH (((V ,d) k) Σ k) kH (((V ,d) k)) Σ k k(H (V ,d) k) Σ k =Sym(H (V ,d)) \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 GVV GV^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.).

References

The symmetric algebra spectrum of the sphere spectrum, and its structure as a Hopf ring spectrum is discussed in

Last revised on December 18, 2017 at 01:13:47. See the history of this page for a list of all contributions to it.