differential graded-commutative algebra



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)deg(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,bAa,b \in A two elements of homogeous degree (n a,σ a),(n b,σ b)×/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) \,.

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

A\phantom{A}Deligne’s conventionA\phantom{A}A\phantom{A}Bernstein’s conventionA\phantom{A}
A\phantom{A}α iα j= \alpha_i \cdot \alpha_j = A\phantom{A}A\phantom{A}(1) (n in j+σ iσ j)α jα i(-1)^{ (n_i \cdot n_j + \sigma_i \cdot \sigma_j) } \alpha_j \cdot \alpha_iA\phantom{A}A\phantom{A}(1) (n i+σ i)(n j+σ j)α jα i (-1)^{ (n_i + \sigma_i) \cdot (n_j + \sigma_j) } \alpha_j \cdot \alpha_iA\phantom{A}
A\phantom{A}common inA\phantom{A}
A\phantom{A}discussion ofA\phantom{A}
A\phantom{A}supergravityA\phantom{A}A\phantom{A}AKSZ sigma-modelsA\phantom{A}
A\phantom{A}Bonora et. al 87,A\phantom{A}
A\phantom{A}Castellani-D’Auria-Fré 91,A\phantom{A}
A\phantom{A}Deligne-Freed 99A\phantom{A}
A\phantom{A}AKSZ 95,A\phantom{A}
A\phantom{A}Carchedi-Roytenberg 12A\phantom{A}

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.


The following are semifree differential graded-commutative algebras:

A\phantom{A}(n,σ)×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}