↑ arvau

Wikipedia / Differential graded algebra

In mathematics – particularly in homological algebra, algebraic topology, and algebraic geometry – a differential graded algebra (or DGA, or DG algebra) is an algebraic structure often used to capture information about a topological or geometric space. Explicitly, a differential graded algebra is a graded associative algebra with a chain complex structure that is compatible with the algebra structure.

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. In algebraic topology, the singular cochains of a topological space form a DGA encoding the singular cohomology. Moreover, American mathematician Dennis Sullivan developed a DGA to encode the rational homotopy type of topological spaces.

Definitions

-graded algebra of degree is a differential graded algebra the structure of a chain complex or cochain complex (depending on the degree), and satisfies a graded Leibniz rule. In what follows, we will denote the "degree" of a homogeneous element satisfies the conditions .

A linear map between graded vector spaces is said to be of degree n. When considering (co)chain complexes, we restrict our attention to chain maps. The morphisms in the category of DGAs are chain maps that are also algebra homomorphisms.

Categorical definition

One can also define DGAs more abstractly using category theory. There is a category of chain complexes over a ring, whose objects are chain complexes and whose morphisms are chain maps. We define the tensor product by

with differential

into a symmetric monoidal category. Then, we can equivalently define a differential graded algebra as a monoid object with an associative and unital multiplication.

Homology and cohomology

is its homology is a subobject. Thus, we can form the quotient

. 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.

). If this induced map is an isomorphism is called a quasi-isomorphism. In many contexts, this is the natural notion of equivalence one uses for (co)chain complexes. We say a morphism of DGAs is a quasi-isomorphism if the chain map on the underlying (co)chain complexes is.

Properties of DGAs

Commutative differential graded algebras

A commutative differential graded algebra (or CDGA, which satisfies a graded version of commutativity. Namely,

. Many of the DGAs commonly encountered in math happen to be CDGAs, like the de Rham algebra of differential forms.

Differential graded Lie algebras

A differential graded Lie algebra (or DGLA) is a differential graded analogue of a Lie algebra, satisfying the following graded analogues of the Lie algebra axioms.

tensored with a Lie algebra, 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.

Formal DGAs

is called formal is formal) that is a quasi-isomorphism. 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.

Examples

Trivial DGAs

Notice that any graded algebra. In particular, as noted above, the (co)homology of any DGA forms a trivial DGA, since it is a graded algebra.

The de-Rham algebra

be a manifold. Then, the differential forms, where the grading is given by form degree. This vector space has a product, given by the exterior product, which makes it into a graded algebra. Finally, the exterior derivative and the graded Leibniz rule. In fact, the exterior product is graded-commutative, which makes the de Rham algebra an example of a CDGA.

Singular cochains

be a topological space its complex of singular cochains with coefficients in a ring, whose cohomology is the singular cohomology, one can define the cup product of cochains, which gives this cochain complex the structure of a DGA., the de Rham theorem states that the singular cohomology is isomorphic to the de Rham cohomology and, moreover, the cup product and exterior product of differential forms induce the same operation on cohomology.

, one can associate a commutative.

The free DGA

be a (non-graded) vector space over a field. The tensor algebra is defined to be the graded algebra

. This is the free algebra.

be any linear map. Then, this extends uniquely to a derivation (homologically graded) by the formula

. This is commonly referred to as the Koszul sign rule.

. The differential is given by

.

The free CDGA

, we define the free graded commutative algebra on it by

denotes the symmetric algebra denotes the exterior algebra is a CDGA in a unique way.

Models for DGAs

is a model if it comes with a surjective that is a quasi-isomorphism.

Minimal models

is minimal if it satisfies the following conditions. be simply connected. mirror the (co)homology groups of a simply connected space.

is a minimal model. The fundamental theorem of minimal models is simply connected then it admits a minimal model, and that if a minimal model exists it is unique up to (non-unique) isomorphism.

The Sullivan minimal model

Minimal models were used with great success by Dennis Sullivan in his work on rational homotopy theory. Given a simplicial complex. is simply connected as a DGA, thus there exists a minimal model.

is a simply connected CW complex.

See also

Notes

References