John Terilla, Thomas Tradler
We develop the deformation theory of A_\infty algebras together with \infty inner products and identify a differential graded Lie algebra that controls the theory. This generalizes the deformation theories of associative algebras, A_\infty algebras, associative algebras with inner products, and A_\infty algebras with inner products.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
The paper has been withdrawn.
We prove that if two associative deformations (parameterized by the same complete local ring) are derived Morita equivalent, then they are Morita equivalent (in the classical sense).
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 aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra $sl_2$ and describe as formal deformations the q-deformed Witt algebra and Jackson $sl_2$.
We compute the second Hochschild cohomology space $HH^2(\mathcal{H}_1)$ of Connes-Moscovici's Hopf algebra $\mathcal{H}_1$, giving the infinitesimal deformations (up to equivalence) of the associative structure. $HH^2(\mathcal{H}_1)$ is shown to be one dimensional, and thus Connes-Moscovici's formal deformation of $\mathcal{H}_1$ using Rankin-Cohen brackets is unique up to equivalence.
We compactify the spaces $K(m,n)$ introduced by Maxim Kontsevich. The initial idea was to construct an $L_\infty$ algebra governing the deformations of a (co)associative bialgebra. However, this compactification leads not to a resolution of the PROP of (co)associative bialgebras, but to a new algebraic structure we call here a CROC. It turns out that these constructions are related to the non-commutative deformations of (co)associative bialgebras. We construct an associative dg algebra conjecturally governing the non-commutative deformations of a bialgebra. Then, using the Quillen duality, we …
We propose a simple approach to formal deformations of associative algebras. It exploits the machinery of multiplicative coresolutions of an associative algebra A in the category of A-bimodules. Specifically, we show that certain first-order deformations of A extend to all orders and we derive explicit recurrent formulas determining this extension. In physical terms, this may be regarded as the deformation quantization of noncommutative Poisson structures on A.
In this paper, we introduce the cohomology theory of $\mathcal{O}$-operators on Hom-associative algebras. This cohomology can also be viewed as the Hochschild cohomology of a certain Hom-associative algebra with coefficients in a suitable bimodule. Next, we study infinitesimal and formal deformations of an $\mathcal{O}$-operator and show that they are governed by the above-defined cohomology. Furthermore, the notion of Nijenhuis elements associated with an $\mathcal{O}$-operator is introduced to characterize trivial infinitesimal deformations.
$G_\infty$-structure is shown to exist on the deformation complex of a morphism of associative algebras. The main step of the construction is extension of a $B_\infty$-algebra by an associative algebra. Actions of $B_\infty$-algebras on associative and $B_\infty$-algebras are analyzed, extensions of $B_\infty$-algebras by associative and $B_\infty$-algebras, that they act upon, are constructed. The resulting $G_\infty$-algebra on the deformation complex of a morphism is shown to be quasi-isomorphic to the $G_\infty$-algebra on deformation complex of the corresponding diagram algebra.
We introduce a concept of $\frac23$PROP generalizing the Kontsevich concept of $\frac12$PROP. We prove that some Stasheff-type compactification of the Kontsevich spaces $K(m,n)$ defines a topological $\frac23$PROP structure. The corresponding chain complex is a minimal model for its cohomology (both are considered as $\frac23$PROPs). We construct a $\frac23$PROP $\End(V)$ for a vector space $V$. Finally, we construct a dg Lie algebra controlling the deformations of a (co)associative bialgebra. Philosophically, this construction is a version of the Markl's operadic construction from [M1] applie…
We introduce triples of associative algebras as a tool for building solutions to the Yang-Baxter equation. It turns out that the class of R-matrices thus obtained is related to a Hecke-like condition, which is formulated for associative algebras with symmetric cyclic inner product. R-matrices for a subclass of the $A_n$-type Belavin-Drinfel'd triples are derived in this way.
The deformation equation and its integrability condition (Bianchi identity) of a non-(co)associative deformation in operad algebra are found. Based on physical analogies, cogravity equations are proposed.
Given an associative multiplication in matrix algebra compatible with the usual one or, in other words, linear deformation of matrix algebra, we construct a solution to the classical Yang-Baxter equation. We also develop a theory of such deformations and construct numerous examples. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures. We also describe an important class of M-structures related to the affine Dynkin diagrams of A, D, E-type. These M-structures and their representations are describe…
In a classic paper, Gerstenhaber showed that first order deformations of an associative k-algebra A are controlled by the second Hochschild cohomology group of A. More generally, any n-parameter first order deformation of A gives, due to commutativity of the cup-product on Hochschild cohomology, a morphism from the graded algebra Sym(k^n) to Ext^*(A,A), the Ext-algebra in the category of A-bimodules. We prove that any extension of the n-parameter first order deformation of A to an INFINITE ORDER formal deformation provides a canonical `lift' of the graded algebra morphism above to a dg-algebra…
In this article, we use Harrison cohomology to provide a framework for commutative deformations. In particular, Kontsevich's result that formality of (the Hochschild complex of) an associative algebra implies its deformability is adapted for commutative algebras, with the Harrison complex.
On deformation theory of associative algebras in monoidal categories
本文通过幺半范畴和余交换余代数推广了结合代数的经典形变理论，构造了函子性范畴框架，证明了形变预层的可表示性，并识别了支配该形变的上同调理论。
Let X be a smooth algebraic variety over a field of characteristic 0. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf O_X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization map from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on equivalence classes.
Classes of $G$-Hom-associative algebras are constructed as deformations of $G$-associative algebras along algebra endomorphisms. As special cases, we obtain Hom-associative and Hom-Lie algebras as deformations of associative and Lie algebras, respectively, along algebra endomorphisms. Chevalley-Eilenberg type homology for Hom-Lie algebras are also constructed.
Let $\{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}}$ be a variational Poisson bracket in a field model on an affine bundle $π$ over an affine base manifold $M^m$. Denote by $\times$ the commutative associative multiplication in the Poisson algebra $\boldsymbol{\mathcal{A}}$ of local functionals $Γ(π)\to\Bbbk$ that take field configurations to numbers. By applying the techniques from geometry of iterated variations, we make well defined the deformation quantization map ${\times}\mapsto{\star}={\times}+\hbar\,\{{\cdot},{\cdot}\}_{\boldsymbol{\mathcal{P}}}+\bar{o}(\hbar)$ that produces a noncommu…
Let X be a smooth algebraic variety over a field K containing the real numbers. We introduce the notion of twisted associative (resp. Poisson) deformation of the structure sheaf of X. These are stack-like versions of usual deformations. We prove that there is a twisted quantization operation from twisted Poisson deformations to twisted associative deformations, which is canonical and bijective on gauge equivalence classes. This result extends work of Kontsevich, and our own earlier work, on deformation quantization of algebraic varieties.
This paper is dedicated to the memory of Moshe Flato, and will appear in Lett. Math. Phys. 48 (1) It became clear during last 5-6 years that the algebraic world of associative algebras (abelian categories, triangulated categories, etc) has many deep connections with the geometric world of two-dimensional surfaces. One of manifestations of this is Deligne's conjecture (1993) which says that on the cohomological Hochschild complex of any associative algebra naturally acts the operad of singular chains in the little discs operad. Recently D. Tamarkin discovered that the operad of chains of the li…
Poisson bialgebras by deformations-to-quasiclassical limits
本文将泊松代数的形变-拟经典极限过程推广到双代数层面，推导了泊松双代数的对应关系并通过相干导子阐释，是李双代数量子化过程的无穷小版本。
We point out that for Yetter's deformational Hochschild complex of a monoidal functor between abelian monoidal categories the Gerstenhaber-Voronov type operations can be defined making it a strong homotopy Gerstenhaber algebra. This encodes deformation theory of monoidal functors in an analogical way as deformation theory of associative algebras is described by the strong homotopy Gerstenhaber algebra structure on the corresponding Hochschild cochains. We describe a quasi-classical limit of deformations of a symmetric monoidal functor in terms of Poisson type structure.
We study non-trivial deformations of the natural action of the Lie algebra $\mathrm{Vect}({\mathbb R}^n)$ on the space of differential forms on ${\mathbb R}^n$. We calculate abstractions for integrability of infinitesimal multi-parameter deformations and determine the commutative associative algebra corresponding to the miniversal deformation in the sense of \cite{ff}.
We define a universal deformation formula (UDF) for the actions of the affine group on Frechet algebras. More precisely, starting with any associative Frechet algebra which the affine group acts on in a strongly continuous and isometrical manner, the UDF produces a family of topological associative algebra structures on the space of smooth vectors of the action deforming the initial product. The deformation field obtained is based over an infinite dimensional parameter space naturally associated with the space of pseudo-differential operators on the real line. This note also presents some geom…
We define a BV-structure on the Hochschild-cohomology of a unital, associative algebra A with a symmetric, invariant and non-degenerate inner product. The induced Gerstenhaber algebra is the one described in Gerstenhaber's original paper on Hochschild-cohomology. We also prove the corresponding theorem in the homotopy case, namely we define the BV-structure on the Hochschild-cohomology of a unital A-infinity-algebra with a symmetric and non-degenerate infinity-inner product.
Solving the Poisson Yang-Baxter equation via deformation-to-quasiclassical-limits
本文将泊松代数由交换结合代数结合形变拟经典极限推导的构造提升至杨-巴克斯特型方程层面，通过标量形变交换结合代数中AYBE解得到PYBE解，引入相关形变概念并借助后泊松代数给出PYBE显式解。
Universal Deformation Formulas (UDFs) for the deformation of associative algebras play a key role in deformation quantization. Here we present examples for certain classes of infinitesimals. A basic representable 2-cocycle $F$ of an associative algebra $\mathcal A$ is one for which there exist commuting derivations $D,\dots, D_n$ of $\mathcal A$ such that $F = \sum_{ij}a_{ij}D_i \smile D_j$, where the $a_{ij}$ are central elements of $\mathcal A$. When $\mathcal A$ is defined over the rationals, there is a natural definition of the exponential of such a cocycle. With this $\exp \hbar F$ define…
Let X be a smooth affine algebraic variety over a field K of characteristic 0, and let R be a complete parameter K-algebra (e.g. R = K[[h]]). We consider associative (resp. Poisson) R-deformations of the structure sheaf O_X. The set of R-deformations has a crossed groupoid (i.e. strict 2-groupoid) structure. Our main result is that there is a canonical equivalence of crossed groupoids from the Deligne crossed groupoid of normalized polydifferential operators (resp. polyderivations) of X to the crossed groupoid of associative (resp. Poisson) R-deformations of O_X. The proof relies on a careful …
One-parameter deformations of the diassociative and dendriform operads
本研究结合Livernet和Loday提出的结合operad极化构造与结合运算的树状分裂方法，借助二次operad的Koszul对偶，实现了非对称树状、双结合operad到对称operad范畴的单参数形变。
The purpose of this paper is to study deformation theory of Hom-associative algebra morphisms and Hom-Lie algebra morphisms. We introduce a suitable cohomology and discuss Infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide some examples.
We study associative multiplications in semi-simple associative algebras over C compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over C. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures in the matrix case and PM-structures in the case of direct sums of several matrix algebras. We also investigate various properties of PM-structures, provide numerous examples and describe an important class of PM-structures. The classification of these PM…
An A-infinity algebra is given by a codifferential on the tensor coalgebra of a (graded) vector space. An associative algebra is a special case of an A-infinity algebra, determined by a quadratic codifferential. The notions of Hochschild and cyclic cohomology generalize from associative to A-infinity algebras, and classify the infinitesimal deformations of the algebra, and those deformations preserving an invariant inner product, respectively. Similarly, an L-infinity algebra is given by a codifferential on the exterior coalgebra of a vector space, with Lie algebras being special cases given b…
This is the text of the lecture given by the author in Naples at "Giornata IndAM", June 7, 2005. The lecture is addressed at the general mathematical audience and reviews several topics in deformation theory of associative algebras.
The classical deformation theory of Lie algebras involves different kinds of Massey products of cohomology classes. Even the condition of extendibility of an infinitesimal deformation to a formal one-parameter deformation of a Lie algebra involves Massey powers of two dimensional cohomology classes which are not powers in the usual definition of Massey products in the cohomology of a differential graded Lie algebra. In the case of deformations with other local bases, one deals with other, more specific Massey products. In the present work a construction of generalized Massey products is given,…
Controlling structures, deformations and homotopy theory for averaging algebras
本文研究结合代数上的相对平均算子与相对平均代数，构造控制其代数结构的分次李代数与 $L_\infty$-代数，定义上同调并研究形变、阿贝尔扩张及同伦相对平均代数。
Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebra $A$ together with a Lie algebra $L$ mapped into the derivations of $A$. A bicomplex (with both Hochschild and Chevalley-Eilenberg cohomologies) is essential.
Covariance of a quantum space with respect to a quantum enveloping algebra ties the deformation of the multiplication of the space algebra to the deformation of the coproduct of the enveloping algebra. Since the deformation of the coproduct is governed by a Drinfeld twist, the same twist naturally defines a covariant star product on the commutative space. However, this product is in general not associative and does not yield the quantum space. It is shown that there are certain Drinfeld twists which realize the associative product of the quantum plane, quantum Euclidean 4-space, and quantum Mi…
The aim of this paper is to review the deformation theory of $n$-Lie algebras. We summarize the 1-parameter formal deformation theory and provide a generalized approach using any unital commutative associative algebra as a deformation base. Moreover, we discuss degenerations and quantization of $n$-Lie algebras.
Given an associative graded algebra equipped with a degree +1 differential we define an A-infinity structure that measures the failure of the differential to be a derivation. This can be seen as a non-commutative analog of generalized BV-algebras. In that spirit we introduce a notion of associative order for the differential and prove that it satisfies properties similar to the commutative case. In particular when it has associative order 2 the new product is a strictly associative product of degree +1 and there is a compatibility between the products, similar to ordinary BV-algebras. We consi…
Deformations of the Lie algebras so(4), so(3,1), and e(3) that leave their so(3) subalgebra undeformed and preserve their coset structure are considered. It is shown that such deformed algebras are associative for any choice of the deformation parameters. Their Casimir operators are obtained and some of their unitary irreducible representations are constructed. For vanishing deformation, the latter go over into those of the corresponding Lie algebras that contain each of the so(3) unitary irreducible representations at most once. It is also proved that similar deformations of the Lie algebras …
We call a finite-dimensional complex Lie algebra $\mathfrak{g}$ strongly rigid if its universal enveloping algebra $\Ug$ is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation. In quantum group theory this phenomenon is well-known to be the case for all complex semisimple Lie algebras. We show that a strongly rigid Lie algebra has to be rigid as Lie algebra, and that in addition its second scalar cohomology group has to vanish (which excludes nilpotent Lie algebras of dimension greater or equal than two). Moreover, using Kontsevit…
The notion of a $F$-manifold algebras is an algebraic description of a $F$-manifold. In this paper, we introduce the notion of Hom-$F$-manifold algebras which is generalisation of $F$-manifold algebras and Hom-Poisson algebras. We develop the representation theory of Hom-$F$-manifold algebras and generalize the notion of Hom-pre-Poisson algebras by introducing the Hom-pre-$F$-manifold algebras which give rise to a Hom-$F$-manifold algebra through the sub-adjacent commutative Hom-associative algebra and the sub-adjacent Hom-Lie algebra. Using $\huaO$-operators on a Hom-$F$-manifold algebras we …
An affine action of an associative algebra $A$ on a vector space $V$ is an algebra morphism $A \to V \rtimes {\rm End}(V)$, where $V$ is a vector space and $V \rtimes {\rm End}(V)$ is the algebra of affine transformations of $V$. The one dimensional version of the Swiss-Cheese operad, denoted ${\mathrm{\bf{sc}}}_1$, is the operad that governs affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by $({\mathrm{\bf{sc}}}_1)_\infty$. Algebras over this minimal model are called Homotopy Affine Actions, they consist of an $A_\infty$-morphism $A \to V \rtim…
In these notes, we introduce formal hom-associative deformations of the quantum planes and the universal enveloping algebras of the two-dimensional non-abelian Lie algebras. We then show that these deformations induce formal hom-Lie deformations of the corresponding Lie algebras constructed by using the commutator as bracket.
An associative algebra is nothing but an odd quadratic codifferential on the tensor coalgebra of a vector space, and an A-infinity algebra is simply an arbitrary odd codifferential. Hochschild cohomology classifies the deformations of an associative algebra into an A-infinity algebra, and cyclic cohomology in the presence of an invariant inner product classifies the deformations of the associative algebra into an A-infinity algebra preserving the inner product. Similarly, a graded Lie algebra is simply a special case of an odd codifferential on the exterior coalgebra of a vector space, and an …
Quantizations of transposed Poisson algebras by Novikov deformations
该研究引入交换结合代数的Novikov形变及经典极限概念，证明其为转置Poisson代数的量子化，推导得Novikov-Poisson型转置Poisson代数均可量子化，并完成二维非阿贝尔复转置Poisson代数量子化的等价分类。
We consider inner deformations of families of $A_\infty$-algebras. With the help of noncommutative Cartan's calculus, we prove the invariance of Hochschild (co)homology under inner deformations. The invariance also holds for cyclic cohomology classes that satisfy some additional conditions. Applications to dg-algebras and QFT problems are briefly discussed.
A simple method is proposed for deforming $A_\infty$-algebras by means of the resolution technique. The method is then applied to the associative algebras of polynomial functions on quantum superspaces. Specifically, by introducing suitable resolutions, we construct explicit deformations of these algebras in the category of minimal $A_\infty$-algebras. The relation of these deformations to higher spin gravities is briefly discussed.
In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven by to compatible with the concepts of homotopy theory: $\infty$-morphisms and the Homotopy Transfer Theorem. We give …
We present a new approach to cyclic homology that does not involve the Connes differential and is based on a `noncommutative equivariant de Rham complex' of an associative algebra. The differential in that complex is a sum of the Karoubi-de Rham differential, which replaces the Connes differential, and another operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit construction of the Gauss-Manin connection, introduced earlier by E. Getzler, on the relative cyclic homology of a flat family of associative algebras over a central base ring. We introduce …
In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and describe some constructions of them. Then we study the formal distribution (noncommutative) Poisson algebras and coefficient (noncommutative) Poisson algebras. Next, we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations.
Non-commutativity and non-associativity are quite natural in string theory. For open strings it appear due to the presence of non-vanishing background two-form in the world volume of Dirichlet brane, while in closed string theory the flux compactifications with non-vanishing three-form also lead to non-geometric backgrounds. In this paper, working in the framework of deformation quantization, we study the violation of associativity imposing the condition that the associator of three elements should vanish whenever each two of them are equal. The corresponding star products are called alternati…
In a deformation quantization of $\Real^n$, the Jacobi identity is automatically satisfied. This article poses the contrary question: Given a set of commutators which satisfies the Jacobi identity, is the resulting associative algebra a deformation quantization of $\Real^n$? It is shown that the result is true. However care must taken when stating precisely how and in which algebra the Jacobi identity is satisfied.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。