Paper computes Atiyah class for DG manifolds of amplitude +1.
problem Computing the Atiyah class for DG manifolds of specific amplitude.
method Computed the Atiyah class by encoding the derived intersection of sections and zero sections of vector bundles.
result Atiyah class vanishes if and only if the intersection is clean.
We study dg-manifolds which are R[2]-bundles over R[1]-bundles over manifolds, we calculate its symmetries, its derived symmetries and we introduce the concept of T-dual dg-manifolds. Within this framework we construct the T-duality map as a degree -1 map between the cohomologies of the T-dual dg-manifolds and we show …
Atiyah classes of DG manifolds of positive amplitude are invariant under weak equivalences.
problem Defining and studying Hochschild cohomology of DG manifolds of positive amplitude.
method Using poly-differential operators and derived intersection, proving invariance under weak equivalences.
result Hochschild cohomology of DG manifolds of positive amplitude is invariant under weak equivalences.
The paper studies formal geometry of dg manifolds and proves isomorphism of their calculi.
problem Formal geometry of dg manifolds.
method Construction of Fedosov dg foliation and homotopy contractions.
result Isomorphism of Cartan and noncommutative calculi.
Unique vertical isomorphisms between Fedosov dg manifolds are proven for Lie pairs.
problem Vertical isomorphisms of Fedosov dg manifolds associated with Lie pairs.
method Construction of Fedosov dg manifolds via splitting and connection, proving unique isomorphisms using iteration formula.
result Existence and uniqueness of vertical isomorphisms between Fedosov dg manifolds.
The paper explores connections between dg manifolds and homotopy Lie algebras.
problem Understanding the relationship between dg manifolds and homotopy Lie algebras.
method Study of formal exponential maps, Atiyah classes, and Kapranov L-infinity algebras.
result Existence of formal exponential maps linked to vanishing of Atiyah classes.
Study Hochschild cohomology of dg manifolds linked to integrable distributions.
problem Understanding Hochschild cohomology of dg manifolds associated with integrable distributions.
method Analyzing the Hochschild cohomology of (F[1],dF) and relating it to the algebra of functions on leaf space. result Established a canonical isomorphism between the Hochschild cohomology of (F[1],dF) and the algebra of functions on leaf space. Constructs Fedosov dg manifolds from Lie pairs.
problem Constructing differential graded manifolds from Lie pairs.
method Fedosov iteration method and homological perturbation lemma.
result Differential graded algebras of functions on the dg manifolds are homotopy equivalent.
Curved L∞ spaces form a category of fibrant objects.
problem Understanding the structure of curved L∞ spaces. method Proving L∞ spaces over dg manifolds form a category of fibrant objects. result Transitive L∞ algebroids over dg manifolds also form a category of fibrant objects. The paper studies Atiyah and Todd classes for DG manifolds derived from integrable distributions.
problem Understanding Atiyah and Todd classes for DG manifolds.
method Analyzing DG manifolds (F[1],dF) corresponding to integrable distributions F. result Atiyah and Todd classes of DG manifolds are identical to those of Lie pairs $(T_{\mathbb{K}} M, F).
Develops L∞ spaces over dg manifolds and establishes an equivalence with L∞ algebroids.
problem Defining and comparing L∞ spaces and algebroids over dg manifolds. method Establishes an equivalence between categories of L∞ algebroids and L∞ spaces, constructs a faithful functor. result Detects weak equivalences between L∞ algebroids and L∞ spaces. Generalizes L-infinity spaces to sheaves over nilpotent dg manifolds.
problem Formal moduli problems and L-infinity spaces.
method Extending L-infinity spaces to sheaves of L-infinity algebras over nilpotent dg manifolds.
result Characteristic classes of the new L-infinity space recover primary invariants of Lie algebroids.
This paper studies Hopf algebras from dg manifolds.
problem Understanding Hopf algebras from the perspective of dg manifolds.
method Analyzes the universal enveloping algebra of Lie algebra objects in homotopy categories of dg modules.
result The universal enveloping algebra of the Lie algebra object is a Hopf algebra.
The paper proves a category of dg manifolds with finite positive amplitude.
problem Understanding the structure of dg manifolds with finite positive amplitude.
method Using path spaces and homotopy transfer theorem for curved L∞[1]-algebras. result Proves that dg manifolds of finite positive amplitude form a category of fibrant objects.
Letters discuss results on Courant algebroids, including classification and reduction.
problem Understanding and classifying Courant algebroids.
method Analyzes properties of Courant algebroids, including exact and transitive ones, and describes them in terms of symplectic dg manifolds.
result Provides a canonical generating Dirac operator and relates CAs to Poisson-Lie T-duality.
Extends Lie bialgebroids to homotopy theory.
problem Characterize Lie bialgebroids and their morphisms.
method Interprets Lie bialgebroids in terms of odd symplectic dg-manifolds.
result Introduces L∞-bialgebroids and their morphisms. The paper constructs structures for Lie pairs and their Atiyah classes.
problem Understanding structures of Lie pairs and their Atiyah classes.
method Constructs dg-manifolds and dg-Lie algebroids for Lie pairs, showing quasi-isomorphisms and Atiyah classes.
result Induces a quasi-isomorphism between dg-Lie algebroids and Atiyah classes of Lie pairs.
We find a minimal differential graded (dg) operad whose generic representations in Rn are in one-to-one correspondence with formal germs of those endomorphisms of the tangent bundle to Rn which satisfy the Nijenhuis integrability condition. This operad is of a surprisingly simple origin -- it is the cobar constru…
We prove that to every inclusion A↪L of Lie algebroids over the same base manifold M corresponds a Kapranov dg-manifold structure on A[1]⊕L/A, which is canonical up to isomorphism. As a consequence, Γ(Λ∙A∨⊗L/A) carries a canonical L∞[1] algebra structure whose una…
Atiyah and Todd classes of Lie algebroids respect their Atiyah sequence.
problem Understanding the Atiyah and Todd classes of Lie algebroids.
method Analyzing the Atiyah sequence of Lie algebroids and proving class restrictions.
result Atiyah and Todd classes of dg manifolds arising from regular Lie algebroids respect the Atiyah sequence.
Study connections on Lie and Courant algebroids, defining basic curvature and Atiyah cocycle.
problem Understanding connections on Lie and Courant algebroids and their compatibility.
method Revisit and define basic curvature for Lie algebroids, introduce basic curvature for Courant algebroids, and use Atiyah cocycle for gauge theory.
result Basic curvature tensor for Courant algebroids and its relation to the Atiyah cocycle.
This paper proves equivalence between derived manifolds and differential graded manifolds.
problem Characterizing derived manifolds and their relationship to differential graded manifolds.
method Proving equivalence between the infinity categories of derived manifolds and differential graded manifolds.
result The infinity category of differential graded manifolds is equivalent to that of derived manifolds.
We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field Q admits a structure of L-infinity algebra with the Lie derivative LQ as unary …
The paper studies deformations of Nijenhuis structures in Lie algebras and algebroids.
problem Deformations of Nijenhuis structures in Lie algebras and algebroids.
method Operadic study, introduction of homotopy Nijenhuis Lie algebras, construction of L∞-algebras for deformations. result The Poincaré Lemma holds for certain Nijenhuis operators, confirming a conjecture.
Standard cohomology of Courant algebroids identified via minimal models.
problem Cohomology of Courant algebroids.
method Minimal model construction and Hodge-to-de Rham spectral sequence.
result Standard cohomology of Courant algebroids identified with function space cohomology.
Develops derived differential geometry theory.
problem Homotopy and intersection in smooth manifolds.
method Using L∞[1]-algebras and homotopy transfer. result Derived manifolds form a category of fibrant objects.
Chern-Weil theory provides for each invariant polynomial on a Lie algebra g a map from g-connections to differential cocycles whose volume holonomy is the corresponding Chern-Simons theory action functional. Kotov and Strobl have observed that this naturally generalizes from Lie algebras to dg-manifolds and dg-bundles …
Let G be a Lie group acting by diffeomorphisms on a manifold M and consider the image of T[1]G and T[1]M, of G and M respectively, in the category of differential graded manifolds. We show that the obstruction to lift the action of T[1]G on T[1]M to an action on a R[n]-bundle over T[1]M is measured by the G equivariant…
New algebraic structure derived from Kähler manifolds.
problem Understanding algebraic structures on differential forms.
method Introducing L∞[1] R-algebras and proving linearization theorems. result Induced L∞[1] R-algebra structures on Γ(L) are linearizable under certain conditions.