symmetric monoidal (∞,1)-category of spectra
A differential graded-commutative algebra (also DGCA or dgca, for short) is a differential-graded algebra which is supercommutative in that for any two elements in homogeneous degree , respectively, then the product in the algebra satisfies
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.
More generally, a differential graded commutative superalgebra 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 and (this Prop.). While equivalent (this Prop.) these yield two superficially different sign rules for differential graded-commutative superalgebras:
for two elements of homogeous degree , respectively, we have
in Deligne’s convention
in Berstein’s convention
While in both cases the differential satisfies.
sign rule for differential graded-commutative superalgebras
(different but equivalent)
Deligne’s convention | Bernstein’s convention | |
---|---|---|
common in discussion of | supergravity | AKSZ sigma-models |
representative references | Bonora et. al 87, Castellani-D’Auria-Fré 91, Deligne-Freed 99 | AKSZ 95, Carchedi-Roytenberg 12 |
Restricted tro bidegree both of these sign rules yield a commutative superalgebra, which restricted to thy yield a differential graded-commutative algebra.
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:
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.
bi-degree | arbitrary | |
commutative algebra | differential graded-commutative algebra | |
arbitrary | e.g. Grassmann algebra | differential graded-commutative superalgebra |
Last revised on September 25, 2020 at 14:57:08. See the history of this page for a list of all contributions to it.