Strong homotopy properads

Homotopy Batalin-Vilkovisky algebras

This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy Batalin-Vilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defind by quadratic and linear relations. The operad encoding Batalin-Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincare-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution …

同伦代数中的普适构造

Universal constructions in homotopical algebra

本文利用李图代数有效积分理论，为多种双代数类型的形变与同伦理论提供普适构造，给出Deligne群胚的同伦描述并推广了operadic代数形变公式。

平衡无穷小双代数、双重 Poisson 代数与预 Calabi-Yau 代数

Balanced infinitesimal bialgebras, double Poisson gebras and pre-Calabi-Yau algebras

本文研究平衡无穷小双代数的 properad，通过 Koszul 对偶与余棒构造揭示其同伦双重 Poisson 代数和预 Calabi-Yau 代数之间的结构关系，并证明弯曲版本等价。

Quantizations of Lie bialgebras, duality involution and oriented graph complexes

We prove that the action of the Grothendieck-Teichmüller group on the genus completed properad of (homotopy) Lie bialgebras commutes with the reversing directions involution of the latter. We also prove that every universal quantization of Lie bialgebras is homotopy equivalent to the one which commutes with the duality involution exchanging Lie bracket and Lie cobracket. The proofs are based on a new result in the theory of oriented graph complexes (which can be of independent interest) saying that the involution on an oriented graph complex that changes all directions on edges induces the ide…

Pre-Calabi--Yau 代数与同伦 double Poisson gebra

Pre-Calabi--Yau algebras and homotopy double Poisson gebras

本文证明了弯曲 pre-Calabi--Yau 代数与弯曲同伦 double Poisson gebra 概念等价，通过证明控制变形理论的微分分次 Lie 代数同构，并应用 properadic calculus 建立了该代数的同伦性质。

Coloured Koszul duality and strongly homotopy operads

This paper proves Koszul duality for coloured operads and uses it to introduce strongly homotopy operads as a suitable homotopy invariant version of operads. It shows that rational chains on configuration spaces of points in the plane form a strongly homotopy operad quasi isomorphic to the chains on the little disks operad.

Chicken or egg? A hierarchy of homotopy algebras

We start by clarifying and extending the multibraces notation, which economically describes substitutions of multilinear maps and tensor products of vectors. We give definitions and examples of homotopy algebras, strongly homotopy Gerstenhaber and Gerstenhaber bracket algebras, and strongly homotopy Batalin-Vilkovisky algebras. We show that a homotopy algebra structure on a vector space can be lifted to its Hochschild complex, and also suggest an induction method to generate explicit strongly homotopy Gerstenhaber algebra maps on a topological vertex operator algebra (TVOA), their existence ha…

Higher derived brackets, strong homotopy associative algebras and Loday pairs

We give a quick method of constructing strong homotopy associative algebra, namely, the higher derived product construction. This method is associative analogue of classical higher derived bracket construction in the category of Loday algebras. We introduce a new type of algebra "Loday pair" which is noncommutative analogue of classical Leibniz pair. We study strong homotopy Loday pairs and the higher derived brackets on the Loday pairs.

Strongly homotopy Lie bialgebras and Lie quasi-bialgebras

Structures of Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasibialgebras are presented as solutions of Maurer-Cartan equations on corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann-Schwarzbach. This approach provides a definition of an $L_\infty$-(quasi)bialgebra (strongly homotopy Lie (quasi)bialgebra). We recover an $L_\infty$-algebra structure as a particular case of our construction. The formal geometry interpretation leads to a definition of an $L_\infty$ (quasi)bialgebra structure on $V$ as a differential operator $Q$ on $V,$ …

Representations of Homotopy Lie-Rinehart Algebras

I propose a definition of left/right connection along a strong homotopy Lie-Rinehart algebra. This allows me to generalize simultaneously representations up to homotopy of Lie algebroids and actions of strong homotopy Lie algebras on graded manifolds. I also discuss the Schouten-Nijenhuis calculus associated to strong homotopy Lie-Rinehart connections.

Courant Algebroids and Strongly Homotopy Lie Algebras

Courant algebroids are structures which include as examples the doubles of Lie bialgebras and the direct sum of tangent and cotangent bundles with the bracket introduced by T. Courant for the study of Dirac structures. Within the category of Courant algebroids one can construct the doubles of Lie bialgebroids, the infinitesimal objects for Poisson groupoids. We show that Courant algebroids can be considered as strongly homotopy Lie algebras.

A Class of Strongly Homotopy Lie Algebras with Simplified sh-Lie Structures

It is known that a single mapping defined on one term of a differential graded vector space extends to a strongly homotopy Lie algebra structure on the graded space when that mapping satisfies two conditions. This strongly homotopy Lie algebra is nontrivial (it is not a Lie algebra); however we show that one can obtain an sh-Lie algebra where the only nonzero mappings defining it are the lower order mappings. This structure applies to a significant class of examples. Moreover in this case the graded space can be replaced by another graded space, with only three nonzero terms, on which the same…

BFV-complex and higher homotopy structures

We present a connection between the BFV-complex (abbreviation for Batalin-Fradkin-Vilkovisky complex) and the so-called strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are $L_{\infty}$ quasi-isomorphic and control the same formal deformation problem. However there is a gap between the non-formal information encoded in the BFV-complex and in the strong hom…

手征高自旋引力的强同伦代数：基于斯托克斯定理

Strong Homotopy Algebras for Chiral Higher Spin Gravity via Stokes Theorem

本文通过斯托克斯定理证明预卡拉比-丘型强同伦代数满足$A_\infty$关系，为手征高自旋引力提供新表述，并暗示统一已知形式性定理的构造。

BRST约化的强同伦结构

The Strong Homotopy Structure of BRST Reduction

本文提出基于$L_\infty$-态射的多微分算子约化方案，利用显式同伦传递定理构造约化态射，并证明其诱导的约化星积与形式Koszul复形所得星积等价。

On the Strong Homotopy Associative Algebra of a Foliation

An involutive distribution $C$ on a smooth manifold $M$ is a Lie-algebroid acting on sections of the normal bundle $TM/C$. It is known that the Chevalley-Eilenberg complex associated to this representation of $C$ possesses the structure $\mathbb{X}$ of a strong homotopy Lie-Rinehart algebra. It is natural to interpret $\mathbb{X}$ as the (derived) Lie-Rinehart algebra of vector fields on the space $\mathbb{P}$ of integral manifolds of $C$. In this paper, I show that $\mathbb{X}$ is embedded in a strong homotopy associative algebra $\mathbb{D}$ of (normal) differential operators. It is natural …

On the Strong Homotopy Lie-Rinehart Algebra of a Foliation

It is well known that a foliation F of a smooth manifold M gives rise to a rich cohomological theory, its characteristic (i.e., leafwise) cohomology. Characteristic cohomologies of F may be interpreted, to some extent, as functions on the space P of integral manifolds (of any dimension) of the characteristic distribution C of F. Similarly, characteristic cohomologies with local coefficients in the normal bundle TM/C of F may be interpreted as vector fields on P. In particular, they possess a (graded) Lie bracket and act on characteristic cohomology H. In this paper, I discuss how both the Lie …

Deformations of coisotropic submanifolds and strong homotopy Lie algebroids

In this paper, we study deformations of coisotropic submanifolds in a symplectic manifold. First we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. This is a non-commutative and non-linear generalization of the well-known description of the local deformation space of Lagrangian submanifolds as the set of graphs of {\it closed} one forms in the Darboux-Weinstein chart of a given Lagrangian submanifold. We then introduce the notion of {\it strong…

来自强同伦代数的非交换规范对称性

Non-commutative gauge symmetry from strong homotopy algebras

本文显式构造了一个 L$_\infty$ 代数，用于在具有任意非交换与非结合星积的空间上定义 U$_\star(1)$ 规范变换，将物质场作为 L$_\infty$ 模纳入其中，并探讨了包含 P$_\infty$ 代数的可能性。

Formal deformations of morphisms of associative algebras

The strong homotopy Lie algebra, controlling simultaneous deformations of a morphism of associative algebras and its domain and codomain is constructed. Isomorphism of the cohomology of this strong homotopy Lie algebra with the classical construction is shown.

The Strong Homotopy Structure of Poisson Reduction

In this paper we propose a reduction scheme for multivector fields phrased in terms of $L_\infty$-morphisms. Using well-know geometric properties of the reduced manifolds we perform a Taylor expansion of multivector fields, which allows us to built up a suitable deformation retract of DGLA's. We first obtained an explicit formula for the $L_\infty$-Projection and -Inclusion of generic DGLA retracts. We then applied this formula to the deformation retract that we constructed in the case of multivector fields on reduced manifolds. This allows us to obtain the desired reduction $L_\infty$-morphis…

Multi-oriented props and homotopy algebras with branes

We introduce a new category of differential graded multi-oriented props whose representations (called homotopy algebras with branes) in a graded vector space require a choice of a collection of $k$ linear subspaces in that space, $k$ being the number of extra directions (if $k=0$ this structure recovers an ordinary prop); symplectic vector spaces equipped with $k$ Lagrangian subspaces play a distinguished role in this theory. Manin triples is a classical example of an algebraic structure (concretely, a Lie bialgebra structure) given in terms of a vector space and its subspace; in the context o…