differential graded-commutative algebra in nLab

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Definition

Differential graded-commutative algebra

A differential graded-commutative algebra (also DGCA or dgca, for short) is a differential-graded algebra which is supercommutative in that for v,wv,w any two elements in homogeneous degree deg(v),deg(w)∈ℤdeg(v), deg(w) \in \mathbb{Z}, respectively, then the product in the algebra satisfies

vw=(−1) deg(v)deg(w)wv. v w \;=\; (-1)^{deg(v) deg(w)} w v \,.

Equivalently this is a commutative monoid in the symmetric monoidal category of chain complexes of vector spaces equipped with the tensor product of chain complexes.

Differential graded-commutative superalgebra

More generally, a differential graded commutative superalgebra (A,d)∈dgcSAlg(A,d) \in dgcSAlg is a commutative monoid in the symmetric monoidal category of chain complexes of super vector spaces.

There are (at least) two such symmetric monoidal structures τ Deligne\tau_{Deligne} and τ Bernst\tau_{Bernst} (this Prop.). While equivalent (this Prop.) these yield two superficially different sign rules for differential graded-commutative superalgebras:

1. for a,b∈Aa,b \in A two elements of homogeous degree (n a,σ a),(n b,σ b)∈ℤ×ℤ/2(n_a, \sigma_a), (n_b, \sigma_b) \in \mathbb{Z} \times \mathbb{Z}/2, respectively, we have

2. in Deligne’s convention

   ab=(−1) n an b+σ aσ bbaa b = (-1)^{n_a n_b + \sigma_a \sigma_b} \, b a

3. in Berstein’s convention

   ab=(−1) (n a+σ a)(n b+σ b)baa b = (-1)^{ (n_a + \sigma_a)(n_b + \sigma_b) } \, b a

While in both cases the differential satisfies.

d(ab)=(da)b+(−1) n 1a(db). d (a b) = (d a) b + (-1)^{n_1} a (d b) \,.

Restricted tro bidegree (0,−)(0,-) both of these sign rules yield a commutative superalgebra, which restricted to (−,even)(-,even) thy yield a differential graded-commutative algebra.

Examples

• The de Rham algebra of differential forms on a smooth manifold is a differential-graded commutative algebra. The algebra of super differential forms on a supermanifold is a differential-graded commutative superalgebra.

• The dg-algebra of polynomial differential forms on an n-simplex;

The following are semifree differential graded-commutative algebras:

• A Sullivan algebra.

• The Chevalley-Eilenberg algebra of a Lie algebra or more generally of an L-infinity algebra or L-infinity algebroid is a differential-graded-commutative algebra, that of a super L-infinity algebra is a differential graded-commutative superalgebra.

A\phantom{A}bi-degreeA\phantom{A} A\phantom{A}(n,σ)∈ℤ×𝔽 2(n,\sigma) \in \mathbb{Z} \times \mathbb{F}_2A\phantom{A}	A\phantom{A}n=0n = 0A\phantom{A}	A\phantom{A}nn\; arbitraryA\phantom{A}
A\phantom{A}σ=even\sigma = evenA\phantom{A}	A\phantom{A}commutative algebraA\phantom{A}	A\phantom{A}differential graded-commutative algebraA\phantom{A}
A\phantom{A}σ\sigma\; arbitraryA\phantom{A}	A\phantom{A}e.g. Grassmann algebraA\phantom{A}	A\phantom{A}differential graded-commutative superalgebraA\phantom{A}

• category of dgc-algebras

• dgc-coalgebra

• model structure on differential graded-commutative algebras

• model structure on differential graded-commutative superalgebras

• rational homotopy theory

• monoidal Dold-Kan correspondence

• geometry of physics – superalgebra – Z-graded commutative superalgebra

• equivariant dgc-algebra

• model structure on equivariant dgc-algebras

• equivariant rational homotopy theory