In geometry, the de Rham algebra of differential forms on a manifold has the structure of a differential graded algebra, and it encodes the de Rham cohomology of the manifold.
Moreover, American mathematician Dennis Sullivan developed a DGA to encode the rational homotopy type of topological spaces.
satisfies the conditions[2] Often one omits the differential and multiplication and simply writes
The morphisms in the category of DGAs are chain maps that are also algebra homomorphisms.
Then, we can equivalently define a differential graded algebra as a monoid object in
th homology group, and all together they form a graded vector space
In fact, the homology groups form a DGA with zero differential.
Analogously, one can define the cohomology groups of a cochain complex, which also form a graded algebra with zero differential.
In many contexts, this is the natural notion of equivalence one uses for (co)chain complexes.
Many of the DGAs commonly encountered in math happen to be CDGAs, like the de Rham algebra of differential forms.
, satisfying the following graded analogues of the Lie algebra axioms.
, with the bracket given by the exterior product of the differential forms and Lie bracket; elements of this DGLA are known as Lie algebra–valued differential forms.
DGLAs also arise frequently in the study of deformations of algebraic structures where, over a field of characteristic 0, "nice" deformation problems are described by the space of Maurer-Cartan elements of some suitable DGLA.
[5] This notion is important, for instance, when one wants to consider quasi-isomorphic chain complexes or DGAs as being equivalent, as in the derived category.
In fact, the exterior product is graded-commutative, which makes the de Rham algebra an example of a CDGA.
Note, however, that while the cup product induces a graded-commutative operation on cohomology, it is not graded commutative directly on cochains.
This vector space can be made into a graded algebra with the multiplication
(homologically graded) by the formula[8] One can think of the minus signs on the right-hand side as coming from "jumping" the map
One can extend this construction to differential graded vector spaces.
is no longer graded by the number of tensor products but instead by the sum of the degrees of the elements of
[9] Similar to the previous case, one can also construct the free CDGA.
[10] As mentioned previously, oftentimes one is most interested in the (co)homology of a DGA.
As such, the specific (co)chain complex we use is less important, as long as it has the right (co)homology.
[11] Since one could form arbitrarily large (co)chain complexes with the same cohomology, it is useful to consider the "smallest" possible model of a DGA.
[12] Note that some conventions, often used in algebraic topology, additionally require that
mirror the (co)homology groups of a simply connected space.
[13] Minimal models were used with great success by Dennis Sullivan in his work on rational homotopy theory.
, one can define a rational analogue of the (real) de Rham algebra: the DGA
is simply connected as a DGA, thus there exists a minimal model.
is a simply connected CW complex with finite dimensional rational homology groups, the minimal model of the CDGA