New algebraic structure derived from Hopf algebra and Drinfel'd twist.
problem Developing a new algebraic structure from existing mathematical concepts.
method Extending L∞-algebra to a Hopf algebra, twisting with Drinfel'd twist, and identifying Hopf morphisms and braided morphisms. result Braided L∞-algebra is derived from the process. Defines and characterizes operators on Lie ∞-algebras with respect to actions.
problem Characterizing operators on Lie ∞-algebras with respect to actions.
method Using higher derived brackets construction and Maurer-Cartan elements.
result Determines the Lie ∞-algebra controlling the deformation of operators.
Lie-Rinehart algebras over C∞-rings defined and studied.
problem Defining and studying Lie-Rinehart algebras over C∞-rings. method Defining Lie-Rinehart algebras over C∞-rings and showing their relationship with Poisson C∞-rings. result A natural Poisson bracket on the C∞-ring associated with a Lie-Rinehart algebra over a C∞-ring. We study the deformation theory of pre-symplectic structures, i.e. closed two-forms of fixed rank. The main result is a parametrization of nearby deformations of a given pre-symplectic structure in terms of an L∞-algebra, which we call Koszul L∞-algebra. This L∞-algebra is a cousin of the Koszul…
The paper explores new algebraic structures and morphisms in graded settings.
problem Understanding new algebraic structures and morphisms in graded settings.
method Introducing and analyzing L∞-, P∞-, and S∞-algebras, and thick morphisms in a Z2imesZ-graded context. result Shifted S∞-thick morphisms induce L∞-morphisms of shifted S∞-structures. 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. Paper proves Koszul duality for weighted A-infinity algebras.
problem Koszul duality for weighted A-infinity algebras.
method Constructs new box tensor product for weighted A-infinity bimodules and verifies correspondence between maps and bimodules.
result Proves Koszul duality result between weighted A-infinity algebras.
An L∞-algebra is built on symplectic manifold homology.
problem No specific problem stated; focuses on construction of algebra.
method Construction of an L∞-algebra on symplectic manifold homology. result The constructed L∞-algebra naturally projects to a Lie algebra extension. Study Morse theory on loop spaces and Hecke algebras.
problem Morse theory applied to loop spaces and Hecke algebras.
method Defined a Morse-type A∞-algebra and showed equivalence to Heegaard Floer algebras. result Equivalence of based multiloop A∞-algebra to wrapped higher-dimensional Heegaard Floer algebras. New algebra structure derived from Lie pairs.
problem Constructing A∞-algebras from Lie pairs. method Using homotopy equivalence and Lie algebroids.
result Chevalley-Eilenberg cohomology gains an associative algebra structure.
Geometrically deforms L∞ algebras to Lie algebroids, revealing new invariants.
problem Classifying geometric invariants of L∞ algebras arising from vector bundles. method Define geometric deformations of curved L∞ algebras and show they correspond to Lie algebroid structures. result Geometric deformations of L∞ algebras classify new geometric invariants. New L∞ algebra governs deformations of Dirac-Jacobi structures.
problem Deformation theory of Dirac-Jacobi structures.
method Using higher derived brackets and split Courant-Jacobi algebroids, an L∞ algebra is associated with each Dirac-Jacobi structure. result There is a one-to-one correspondence between MC elements of the L∞ algebra and small deformations of the Dirac-Jacobi structure. The paper extends Chern-Weil-Lecomte map to L∞-algebras.
problem Defining characteristic classes for L∞-algebra extensions. method Using the Chern-Weil-Lecomte map to define characteristic classes in an L∞-algebra setting. result Unified definition of several known cohomology classes.
This Master Thesis is devoted to the study of n-plectic manifolds and the Strongly Homotopy Lie algebras, also called L∞-algebras, that can be associated to them. Since multisymplectic geometry and L∞-algebras are relevant in Theoretical Physics, and in particular in String Theory, we introduce th…
This paper upgrades Khovanov homology to an L-infinity module structure.
problem Exploring Khovanov homology with L-infinity algebra structures.
method Developed an L-infinity algebra structure on sl2(∧) and showed annular Khovanov homology is an L-infinity module over it.
result The annular Khovanov homology of a link L is an L-infinity module over sl2(∧) up to quasi-isomorphism.
The settings for homotopical algebra---categories such as simplicial groups, simplicial rings, A∞ spaces, E∞ ring spectra, etc.---are often equivalent to categories of algebras over some monad or triple T. In such cases, T is acting on a nice simplicial model category in such a way that T descends…
The paper develops a theory of C∞-superrings and their superschemes.
problem Developing a theory for C∞-superrings and superschemes. method Proving an equivalence between categories of fair affine C∞-superschemes and fair C∞-superrings. result A key equivalence between fair affine C∞-superschemes and fair C∞-superrings. If X is a smooth manifold then the R-algebra C∞(X) of smooth functions c:X→R is a C∞-ring. That is, for each smooth function f:Rn→R there is an n-fold operation Φf:C∞(X)n→C∞(X) acting by Φf:(c1,…,cn)↦f(c1,...,cn), a…
Lie's third theorem proven for Lie ∞-algebras.
problem Integrating finite-type Lie ∞-algebras to Lie ∞-groups.
method Local minimal models for Kan simplicial manifolds.
result Every finite-type Lie ∞-algebra integrates to a finite-dimensional Lie ∞-group.
Let L be a line bundle over M. In this paper we associate an L∞-algebra to any L-Courant algebroid (contact Courant algebroid in the sense of Grabowski). This construction is similar to the work of Roytenberg and Weinstein for Courant algebroids. Next we associate a p-term L∞-algebra to any isot…
We construct a functor from the derived category of homotopy Gerstenhaber algebras with finite-dimensional cohomology to the purely geometric category of so-called F∞-manifolds. The latter contains Frobenius manifolds as a subcategory (so that a pointed Frobenius manifold is itself a homotopy Gerstenhaber alg…
Let $\g\_2$ be the Hochschild complex of cochains on $C^\infty(\RM^n)$ and $\g\_1$ be the space of multivector fields on $\RM^n$. In this paper we prove that given any G_∞-structure ({\rm i.e.} Gerstenhaber algebra up to homotopy structure) on $\g\_2$, and any C_∞-morphism φ ({\rm i.e.} morphism of co…
We solve higher-order morphisms for twisted Courant algebras.
problem Construct canonical L∞-morphisms for higher Courant algebroids. method Develop a general framework for arbitrary r. result Affirmative answer to Zambon's question for higher degrees.
We show how to integrate a weak morphism of Lie algebra crossed-modules to a weak morphism of Lie 2-groups. To do so we develop a theory of butterflies for 2-term L_infty algebras. In particular, we obtain a new description of the bicategory of 2-term L_infty algebras. We use butterflies to give a functorial constructi…
We investigate Nijenhuis deformations of L∞-algebras, a notion that unifies several Nijenhuis deformations, namely those of Lie algebras, Lie algebroids, Poisson structures and Courant structures. Additional examples, linked to Lie n-algebras and n-plectic manifolds, are included.
We explain how deformation theories of geometric objects such as complex structures, Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson algebras. We use homological perturbation theory to obtain A∞ algebra structures and some canonically defined deformations of s…
I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on ma…
The abstract introduces a new A∞ duality via LSFT algebra.
problem Legendrian knot duality and its A∞ extension. method Using Ng's LSFT algebra, the abstract upgrades duality to a quasi-isomorphism of A∞ bimodules over Aug+. result Explicit construction of homotopy inverse for the A∞ Sabloff map. The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra which depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an L∞ algebra instead. We develop a simplified method for describing this L∞ algebra a…
This paper reinterprets Khovanov-Sano symmetries using BV formalism.
problem Understanding symmetries in equivariant Khovanov homology.
method Identifying Shumakovitch operator as a BV Laplacian and proving L∞-algebra structure. result Construction of an intrinsic L∞-algebra on the Khovanov-Sano complex. This research extends Lie algebra actions to singular foliations.
problem Understanding symmetries in singular foliations without additional assumptions.
method Equivalence of categories between Lie-Rinehart algebras and Lie ∞-algebroids. result Universal Lie ∞-algebroids for singular foliations. If X is a manifold then the set C∞(X) of smooth functions f:X→R is a C∞-ring, a rich algebraic structure with many operations. C∞-schemes are schemes over C∞-rings, a way of using Algebro-Geometric techniques in Differential Geometry. They include smooth manifolds, but also…
Paper studies symmetries in singular foliations using Lie ∞-morphisms.
problem Understanding symmetries in singular foliations.
method Analyzes Lie ∞-morphisms induced by Lie algebra actions on singular foliations. result Deduces geometrical consequences, including examples of non-extendable symmetries.
We describe two constructions giving rise to curved A∞-algebras. The first consists of deforming A∞-algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a …
The L∞-algebra is an algebraic structure suitable for describing deformation problems. In this paper we construct one L∞-algebra, which turns out to be a differential graded Lie algebra, to control the deformations of Lie algebroids and a second one to control the deformations of Lie subalgebroids. We a…
Lie algebroids and curved Lie algebras are equivalent categories.
problem Understanding the relationship between Lie algebroids and curved Lie algebras.
method Developed a method to study the ∞-category of curved Lie algebras using homotopy theory of algebras over a complete operad. result Equivalence of ∞-categories between Lie algebroids and certain kinds of curved Lie algebras. 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.
Algebras of smooth functions help reconstruct bulk topological types.
problem Reconstructing the smooth topological type of a compact manifold from its boundary.
method Introducing subalgebras of boundary functions and proving their tensor product reconstruction of the bulk algebra.
result The topological algebras A(v) and B(f) allow for the recovery of the smooth topological type of the bulk X. In this paper, we attach an L∞-algebra to any coisotropic submanifold in a Jacobi manifold. Our construction generalizes and unifies analogous constructions by Oh-Park (symplectic case), Cattaneo-Felder (Poisson case), Lê-Oh (locally conformal symplectic case). As a new special case, we attach an L∞-alg…
We study Maurer-Cartan elements on homotopy Poisson manifolds of degree n. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson $\g$-manifolds, and twisted Courant algebroids. Using the fact that the dual of an n-term $L_\infty…
Constructs combinatorial 2D topological field theories from cyclic A-infinity algebras.
problem Developing a combinatorial framework for 2D topological field theories.
method Using triangulations and polygonal decompositions, constructing cochains on a CW complex.
result Existence of combinatorial 2D topological field theories based on cyclic A-infinity algebras.
Solves differentiation for Lie ∞-groups using formal groupoids.
problem Differentiation of Lie ∞-groups.
method Develops homotopy theory of formal ∞-groupoids and analyzes Dold-Kan adjunction for cosimplicial algebras.
result Differentiation functor from finite-dimensional Lie ∞-groups to finite-type Lie ∞-algebras is homotopically well-behaved.
Defines new algebras for virtual link invariants, matching known polynomials.
problem Developing new mathematical structures for virtual link invariants.
method Introducing two towers of algebras, VTL and ATL, and determining their presentations and Markov traces.
result The invariants derived from the Markov traces match known polynomials for virtual links.
In this paper we define the p-adic framed braid group F∞,n, arising as the inverse limit of the modular framed braids and we give topological generators for F∞,n. We also give geometric interpretations for the p-adic framed braids. We then construct a p-adic Yokonuma-Hec…
Constructs Koszul dual algebras for star-shaped diagrams in 3-manifolds.
problem Constructing algebraic structures for 3-manifold homology.
method Uses graphical calculus to construct Koszul dual weighted A∞-algebras and dualizing bimodules. result Proves duality of constructed algebras and bimodules.
The paper proves a generalized inverse function theorem for curved L∞ spaces.
problem Proving a generalized inverse function theorem for curved L∞ spaces. method Obstruction theory for L∞ homomorphisms and homotopy transfer theorem for curved L∞ algebras. result A morphism of curved L∞ spaces which is a quasi-isomorphism at a point has a local homotopy inverse. Study derived Lie ∞-groupoids and algebroids in higher differential geometry.
problem Addressing problems in higher differential geometry using derived Lie ∞-groupoids and algebroids.
method Construct CFO structures, study L∞-algebroids, homotopical algebras, and homotopy-coherent representations.
result Construct Atiyah classes for L∞-algebroids pairs and study singular foliations and their holonomies.
This paper defines a functor for L∞-modules and applies it to Khovanov homology.
problem Defining a functor for L∞-modules and proving its properties. method Abstract approach using morphisms of L∞-algebras. result Restriction of scalars defines a functor between L∞-modules.