Frank Schuhmacher
We show that three deformation functors (deformations of the product, flat deformations and deformations of the relations) assigned to an associative algebra are naturally isomorphic.
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The aim of this paper is to establish a contravariant adjunction between the category of quasi-bialgebras and a suitable full subcategory of dual quasi-bialgebras, adapting the notion of finite dual to this framework. Various functorial constructions involving non-associative algebras and non-coassociative coalgebras are then carried out. Several examples illustrating our methods are expounded as well.
We introduce an equivariant version of Hochschild cohomology as the deformation cohomology to study equivariant deformations of associative algebras equipped with finite group actions.
In this paper, we consider compatible Hom-associative algebras as a twisted version of compatible associative algebras. Compatible Hom-associative algebras are characterized as Maurer-Cartan elements in a suitable bidifferential graded Lie algebra. We also define a cohomology theory for compatible Hom-associative algebras generalizing the classical case. As applications of cohomology, we study abelian extensions and deformations of compatible Hom-associative algebras.
Derivations, Cohomology and Deformation of BiHom-Associative Dialgebra
本文定义并分类了2维和3维BiHom-结合双代数的导子、广义导子与拟导子，将BiHom-结合代数的上同调推广到双代数情形，并利用上同调研究了其单参数形式形变。
We study deformation quantization of nonassociative algebras whose associator satisfies some symmetric relations. This study is expanded to a larger class of nonassociative algebras includind Leibniz algebras. We apply also to this class the rule of polarization-depolarization.
Cohomology and deformation theory of Reynolds--Nijenhuis associative algebras
本文提出Reynolds-Nijenhuis交换代数的上同调与变形理论，探讨其与Rota-Baxter算子的关系及变形理论中的关键性质。
Deformation maps of Quasi-twilled associative algebras
本文引入拟扭配对结合代数的两类形变映射，统一多种结合代数算子，并建立其控制代数与上同调，同时获得若干新结果。
Adjoint functors between the categories of crossed modules of dialgebras and Leibniz algebras are constructed. The well-known relations between the categories of Lie, Leibniz, associative algebras and dialgebras are extended to the respective categories of crossed modules.
Non-associative Categories of Octonionic Bimodules
本文研究八元数双模的非结合范畴，通过修改复合运算引入Hom和Tensor函子并建立Yoneda引理，利用包络范畴确立了函子的正合性。
Cohomologies and deformations of weighted Rota-Baxter Lie algebras and associative algebras with derivations
该研究引入带导子的加权Rota-Baxter李代数与结合代数配对概念，构建其上同调并探究关联，还应用于形变研究，丰富了相关代数结构理论。
We study a special class of weakly associative algebras: the symmetric Leibniz algebras. We describe the structure of the commutative and skew symmetric algebras associated with the polarization-depolarization principle. We also give a structure theorem for the symmetric Leibniz algebras and we study formal deformations in the context of deformation quantization.
The aim of this paper is to give an overview and to compare the different deformation theories of algebraic structures. We describe in each case the corresponding notions of degeneration and rigidity. We illustrate these notions with examples and give some general properties. The last part of this work shows how these notions help in the study of associative algebras varieties. The first and popular deformation approach was introduced by M. Gerstenhaber for rings and algebras using formal power series. A noncommutative version was given by Pinczon and generalized by F. Nadaud. A more general a…
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).
We investigate Lie algebras whose Lie bracket is also an associative or cubic associative multiplication to characterize the class of nilpotent Lie algebras with a nilindex equal to 2 or 3. In particular we study the class of 2-step nilpotent Lie algebras, their deformations and we compute the cohomology which parametrize the deformations in this class.
We introduce the first hom-associative Weyl algebras over a field of prime characteristic as a generalization of the first associative Weyl algebra in prime characteristic. First, we study properties of hom-associative algebras constructed from associative algebras by a general "twisting" procedure. Then, with the help of these results, we determine the commuter, center, nuclei, and set of derivations of the first hom-associative Weyl algebras. We also classify them up to isomorphism, and show, among other things, that all nonzero endomorphisms on them are injective, but not surjective. Last, …
Quasi-twilled associative algebras, deformation maps and their governing algebras
本文研究拟扭织结合代数，引入强形变映射和弱形变映射，统一刻画同态、导子、Rota-Baxter 算子等多类算子，并建立其同调与控制代数以研究同时形变。
Algebraic scheme for constructing deformations of structure constants for associative algebras generated by a deformation driving algebras (DDAs) is discussed. An ideal of left divisors of zero plays a central role in this construction. Deformations of associative three-dimensional algebras with the DDA being a three-dimensional Lie algebra and their connection with integrable systems are studied.
Deformations and extensions of homotopy associative algebras
本文发展了结合2-代数的表示与上同调理论，并详细研究了其形变与阿贝尔扩张。
We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as infinitesimal deformations and to solve the equation of deformations in a polynomial frame. We consider also the deformations of the enveloping algebra of a rigid Lie algebra and we define valued deformations for some classes of non associative algebras.
Considering a Poisson algebra as a non associative algebra satisfying the Markl-Remm identity, we study deformations of Poisson algebras as deformations of this non associative algebra. This gives a natural interpretation of deformations which preserves the underlying associative structure and we study deformations which preserve the underlying Lie algebra.
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 $sl_2$.
In this paper, we consider Rota-Baxter operators on involutive associative algebras. We define cohomology for Rota-Baxter operators on involutive algebras that governs the formal deformation of the operator. This cohomology can be seen as the Hochschild cohomology of a certain involutive associative algebra with coefficients in a suitable involutive bimodule. We also relate this cohomology with the cohomology of involutive dendriform algebras. Finally, we show that the standard Fard-Guo construction of the functor from the category of dendriform algebras to the category of Rota-Baxter algebras…
Deformation quantization of nonassociative algebras
本文研究了K[Σ3]-结合代数等非结合代数的形式形变，通过极化/去极化过程确定对应代数结构，证明形式形变即为该结构的量子化形变，并揭示了反结合代数与Jacobi-Jordan代数的联系。
The first (associative) Weyl algebra is formally rigid in the classical sense. In this paper, we show that it can however be formally deformed in a nontrivial way when considered as a so-called hom-associative algebra, and that this deformation preserves properties such as the commuter, while deforming others, such as the center, power associativity, the set of derivations, and some commutation relations. We then show that this deformation induces a formal deformation of the corresponding Lie algebra into what is known as a hom-Lie algebra, when using the commutator as bracket. We also prove t…
In this paper, we study the moduli space of $1|2$-dimensional complex associative algebras, which is also the moduli space of codifferentials on the tensor coalgebra of a $2|1$-dimensional complex space. We construct the moduli space by considering extensions of lower dimensional algebras. We also construct miniversal deformations of these algebras. This gives a complete description of how the moduli space is glued together via jump deformations.
In this paper, we study the moduli space of $2|1$-dimensional complex associative algebras, which is also the moduli space of codifferentials on the tensor coalgebra of a $1|2$-dimensional complex space. We construct the moduli space by considering extensions of lower dimensional algebras. We also construct miniversal deformations of these algebras. This gives a complete description of how the moduli space is glued together via jump deformations.
Hom-dendriform algebras are twisted analog of dendriform algebras and are splitting of hom-associative algebras. In this paper, we define a cohomology and deformation for hom-dendriform algebras. We relate this cohomology with the Hochschild-type cohomology of hom-associative algebras. We also describe similar results for the twisted analog of dendriform coalgebras.
Deformations of the structure constants for a class of associative noncommutative algebras generated by Deformation Driving Algebras (DDA's) are defined and studied. These deformations are governed by the Central System (CS). Such a CS is studied for the case of DDA being the algebra of shifts. Concrete examples of deformations for the three-dimensional algebra governed by discrete and mixed continuous-discrete Boussinesq (BSQ) and WDVV equations are presented. It is shown that the theory of the Darboux transformations, at least for the BSQ case, is completely incorporated into the proposed sc…
Globalization of Partial Group Actions on Not Necessarily Associative Algebras and Covariant Representations
本文研究非结合代数中部分群作用的全球化问题，提出Λ构造方法，并探讨其在李代数协变表示中的应用及范畴间的伴随函子。
Discrete and q-difference deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by a central system of discrete or q-difference equations which in particular cases represent discrete and q-difference versions of the oriented associativity equation. It is demonstrated also that the celebrated Hirota-Miwa bilinear equation for the AKP and BKP hierarchies describes discrete deformations of certain finite-dimensional algebras.
Anti-associative algebras
该研究围绕反结合代数展开，刻画了有限生成自由反结合代数，明确其与Jacobi-Jordan代数等非结合代数的关联，还基于Markl-Remm工作描述了对应代数类的operad及形变相关上同调复形。
Rota-Baxter operators and more generally $\mathcal{O}$-operators on associative algebras are important in probability, combinatorics, associative Yang-Baxter equation and splitting of algebras. Using a method of Uchino, we construct an explicit graded Lie algebra whose Maurer-Cartan elements are given by $\mathcal{O}$-operators. This allows us to construct a cohomology for an $\mathcal{O}$-operator. This cohomology can also be seen as the Hochschild cohomology of a certain algebra with coefficients in a suitable representation. Next, we study linear and formal deformations of an $\mathcal{O}$-…
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.
Interrelations between discrete deformations of the structure constants for associative algebras and discrete integrable systems are reviewed. A theory of deformations for associative algebras is presented. Closed left ideal generated by the elements representing the multiplication table plays a central role in this theory. Deformations of the structure constants are generated by the Deformation Driving Algebra and governed by the central system of equations. It is demonstrated that many discrete equations like discrete Boussinesq equation, discrete WDVV equation, discrete Schwarzian KP and BK…
In this paper, we give a purely cohomological interpretation of the extension problem for associative algebras; that is the problem of extending an associative algebra by another associative algebra. We then give a similar interpretation of infinitesimal deformations of extensions. In particular, we consider infinitesimal deformations of representations of an associative algebra.
We study the degeneration relations on the varieties of associative and Lie algebra structures on a fixed finite dimensional vector space and give a description of them in terms of Gerstenhaber formal deformations. We use this result to show how the orbit closure of the $3$-dimensional Lie algebras can be determined using homological algebra. For the case of finite dimensional associative algebras, we prove that the $N$-Koszul property is preserved under the degeneration relation for all $N\geq2$.
In this paper, we study moduli spaces of 2-dimensional complex associative algebras. We give a complete calculation of the cohomology of every element in the moduli space, as well as compute their versal deformations.
Associative $H$-pseudoalgebras with a semigroup
本文提出由半群索引的$Ω$-结合$H$-伪代数，给出多种构造方法，并建立其上同调理论及其对形式形变的控制，应用于导出$Ω$-Poisson $H$-伪代数。
In this paper, we study the moduli space of $1|1$-dimensional complex associative algebras. We give a complete calculation of the cohomology of every element in the moduli space, as well as compute their versal deformations.
We describe the general non-associative version of Lie theory that relates unital formal multiplications (formal loops), Sabinin algebras and non-associative bialgebras. Starting with a formal multiplication we construct a non-associative bialgebra, namely, the bialgebra of distributions with the convolution product. Considering the primitive elements in this bialgebra gives a functor from formal loops to Sabinin algebras. We compare this functor to that of Mikheev and Sabinin and show that although the brackets given by both constructions coincide, the multioperator does not. We also show how…
For a given extension $A \subset E$ of associative algebras we describe and classify up to an isomorphism all $A$-complements of $E$, i.e. all subalgebras $X$ of $E$ such that $E = A + X$ and $A \cap X = \{0\}$. Let $X$ be a given complement and $(A, \, X, \, \triangleright, \triangleleft, \leftharpoonup, \rightharpoonup \bigl)$ the canonical matched pair associated with the factorization $E = A + X$. We introduce a new type of deformation of the algebra $X$ by means of the given matched pair and prove that all $A$-complements of $E$ are isomorphic to such a deformation of $X$. Several explici…
Quantum deformations of the structure constants for a class of associative noncommutative algebras are studied. It is shown that these deformations are governed by the quantum central systems which has a geometrical meaning of vanishing Riemann curvature tensor for Christoffel symbols identified with the structure constants. A subclass of isoassociative quantum deformations is described by the oriented associativity equation and, in particular, by the WDVV equation. It is demonstrated that a wider class of weakly (non)associative quantum deformations is connected with the integrable soliton eq…
In this paper, we first propose the concept of Rota-Baxter family $Ω$-associative conformal algebras, then we study the cohomology theory of Rota-Baxter family $Ω$-associative conformal algebras of any weight and justify it by interpreting the lower degree cohomology groups as formal deformations.
In this paper we study the moduli space of 4-dimensional complex associative algebras. We use extensions to compute the moduli space, and then give a decomposition of this moduli space into strata consisting of complex projective orbifolds, glued together through jump deformations. Because the space of 4-dimensional algebras is large, we only classify the non-nilpotent algebras in this paper.
A hom-associative algebra is an algebra whose associativity is twisted by an algebra homomorphism. In this paper, we define a cup product on the cohomology of a hom-associative algebra. We show that the cup product together with the degree $-1$ graded Lie bracket (which controls the deformation of the hom-associative algebra structure) on the cohomology forms a Gerstenhaber algebra. This generalizes a classical fact that the Hochschild cohomology of an associative algebra carries a Gerstenhaber algebra structure.
In this review article, first we give the concrete formulas of representations and cohomologies of associative algebras, Lie algebras, pre-Lie algebras, Leibniz algebras and 3-Lie algebras and some of their strong homotopy analogues. Then we recall the graded Lie algebras and graded associative algebras that characterize these algebraic structures as Maurer-Cartan elements. The corresponding Maurer-Cartan element equips the graded Lie or associative algebra with a differential. Then the deformations of the given algebraic structures are characterized as the Maurer-Cartan elements of the result…
We consider an intermediate category between the category of finite quivers and a certain category of pseudocompact associative algebras whose objects include all pointed finite dimensional algebras. We define the completed path algebra and the Gabriel quiver as functors. We give an explicit quotient of the category of algebras on which these functors form an adjoint pair. We show that these functors respect ideals, obtaining in this way an equivalence between related categories.
A construction of deformations to general algebras
本文针对“给定有限维结合C-代数A，找出可形变为A的代数”这一形变理论相关问题，提出生成结合平坦形变的简单方法，并应用该方法解答了Michael Wemyss关于收缩代数形变的问题。
This is an English translation of the author's Ph.D. thesis, accumulating his results on a construction of Cohen-Macaulay modules over a polynomial ring that appeared in the study of Cauchy-Fueter equations. This construction is generalized from quaternions to arbitrary finite-dimensional associative algebras. We show that for maximally central algebras (as introduced by Azumaya) this construction produces Cohen-Macaulay modules and is an exact functor (tensoring with a bimodule, actually) and this class of algebras cannot be enlarged. For this class several invariants of the resulting modules…
Cohomology theory of Nijenhuis family $Ω$-associative algebras
本文引入Nijenhuis族Ω-结合代数概念，建立其上同调理论，并证明该上同调控制形变，进而用第二上同调群刻画其阿贝尔扩张。
Deformations and homotopy theory of Nijenhuis associative algebras
本文引入同伦Nijenhuis结合代数概念，证明其控制算子为极小模型，构造控制结合积与Nijenhuis算子同时形变的 $L_\infty$-代数，并导出形变复形与上同调理论。
Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras ar…
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…
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。
Cohomology of Hom-associative algebras in Loday-Pirashvili category with applications
本文在Loday-Pirashvili范畴中引入Hom-结合代数结构，研究其上同调理论，并应用于形变与交换扩张，证明交换扩张等价类与第二上同调群元素一一对应。
In this paper, we define and study (co)homology theories of a compatible associative algebra $A$. At first, we construct a new graded Lie algebra whose Maurer-Cartan elements are given by compatible associative structures. Then we define the cohomology of a compatible associative algebra $A$ and as applications, we study extensions, deformations and extensibility of finite order deformations of $A$. We end this paper by considering compatible presimplicial vector spaces and the homology of compatible associative algebras.
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.
Flat model structures and Gorenstein objects in functor categories
本文构建了在泛函范畴上的平坦模型结构，并证明了在特定条件下，对象的戈尔登斯坦性质等价于其在每个对象上的性质，改进了已有结论。
Extension, deformation and categorification of $\text{AssDer}$ pairs
本文研究带有导子的结合代数（AssDer pair），通过上同调探讨其中心扩张与形式单参数形变，并证明带 $2$-导子的结合 2-代数范畴等价于带同伦导子的 2-项 $A_\infty$-代数范畴。
In this survey, we first present basic facts on A-infinity algebras and modules including their use in describing triangulated categories. Then we describe the Quillen model approach to A-infinity structures following K. Lefevre's thesis. Finally, starting from an idea of V. Lyubashenko's, we give a conceptual construction of A-infinity functor categories using a suitable closed monoidal category of cocategories. In particular, this yields a natural construction of the bialgebra structure on the bar construction of the Hochschild complex of an associative algebra.
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