nLab differential graded-commutative algebra

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 $v,w$ any two elements in homogeneous degree $deg(v), deg(w) \in \mathbb{Z}$, respectively, then the product in the algebra satisfies

$$v w \;=\; (-1)^{deg(v) deg(w)} w v \,.$$

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 $(A,d) \in dgcSAlg$ 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 $\tau_{Deligne}$ and $\tau_{Bernst}$ (this Prop.). While equivalent (this Prop.) these yield two superficially different sign rules for differential graded-commutative superalgebras:

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

2. in Deligne’s convention

   $a b = (-1)^{n_a n_b + \sigma_a \sigma_b} \, b a$

3. in Berstein’s convention

   $a b = (-1)^{ (n_a + \sigma_a)(n_b + \sigma_b) } \, b a$

While in both cases the differential satisfies.

$$d (a b) = (d a) b + (-1)^{n_1} a (d b) \,.$$

sign rule for differential graded-commutative superalgebras
(different but equivalent)

	$\phantom{A}$Deligne’s convention$\phantom{A}$	$\phantom{A}$Bernstein’s convention$\phantom{A}$
$\phantom{A}$$\alpha_i \cdot \alpha_j =$$\phantom{A}$	$\phantom{A}$$(-1)^{ (n_i \cdot n_j + \sigma_i \cdot \sigma_j) } \alpha_j \cdot \alpha_i$$\phantom{A}$	$\phantom{A}$$(-1)^{ (n_i + \sigma_i) \cdot (n_j + \sigma_j) } \alpha_j \cdot \alpha_i$$\phantom{A}$
$\phantom{A}$common in$\phantom{A}$ $\phantom{A}$discussion of$\phantom{A}$	$\phantom{A}$supergravity$\phantom{A}$	$\phantom{A}$AKSZ sigma-models$\phantom{A}$
$\phantom{A}$representative$\phantom{A}$ $\phantom{A}$references$\phantom{A}$	$\phantom{A}$Bonora et. al 87,$\phantom{A}$ $\phantom{A}$Castellani-D’Auria-Fré 91,$\phantom{A}$ $\phantom{A}$Deligne-Freed 99$\phantom{A}$	$\phantom{A}$AKSZ 95,$\phantom{A}$ $\phantom{A}$Carchedi-Roytenberg 12$\phantom{A}$

Restricted tro bidegree $(0,-)$ both of these sign rules yield a commutative superalgebra, which restricted to $(-,even)$ 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.

$\phantom{A}$bi-degree$\phantom{A}$ $\phantom{A}$$(n,\sigma) \in \mathbb{Z} \times \mathbb{F}_2$$\phantom{A}$	$\phantom{A}$$n = 0$$\phantom{A}$	$\phantom{A}$$n\;$ arbitrary$\phantom{A}$
$\phantom{A}$$\sigma = even$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$	$\phantom{A}$differential graded-commutative algebra$\phantom{A}$
$\phantom{A}$$\sigma\;$ arbitrary$\phantom{A}$	$\phantom{A}$e.g. Grassmann algebra$\phantom{A}$	$\phantom{A}$differential graded-commutative superalgebra$\phantom{A}$

Related concepts

• 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