Infinitesimal deformations of restricted simple Lie algebras II

Simple finite group schemes and their infinitesimal deformations

We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.

Infinity Algebras, Cohomology and Cyclic Cohomology, and Infinitesimal Deformations

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…

Deformations of the restricted Melikian Lie algebra

We compute the infinitesimal deformations of the restricted Melikian Lie algebra in characteristic 5.

Infinitesimal deformations of null-filiform Leibniz superalgebras

In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras $NF^{n,m}$. One of them is a Leibniz algebra (that is $m=0$) and the second one is a pure Leibniz superalgebra (that is $m\neq 0$) of maximum nilindex. We show that the closure of union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable defor…

左对称彩色代数的变形

Deformations of left-symmetric color algebras

本文研究了左对称彩色代数的变形理论，探讨了变形的等价类和扩展性，并分析了Nijenhuis和Rota-Baxter算子对平凡变形的影响。

On some constructions of nil-clean, clean, and exchange rings

In this paper we discuss several constructions that lead to new examples of nil-clean, clean, and exchange rings. A characterization of the idempotents in the algebra defined by a 2-cocycle is given and used to prove some of the algebra's properties (the infinitesimal deformation case). From infinitesimal deformations we go to full deformations and prove that any formal deformation of a clean (exchange) ring is itself clean (exchange). Examples of nil-clean, clean, and exchange rings arising from poset algebras are also discussed.

后李代数与后霍普夫代数对应关系的无穷小形变

An infinitesimal deformation of the post-Lie and post-Hopf algebra correspondence

该研究证明后李与后霍普夫代数的伴随关系与无穷小结构相容，推广了Cartier–Milnor–Moore定理，完成了相关结构的分类并证明了无穷小后李代数operad的Koszul性。

Cohomology and deformations of 3-dimensional Heisenberg Hom-Lie superalgebras

In this paper, we study Hom-Lie superalgebras of Heisenberg type. For 3-dimensional Heisenberg Hom-Lie superalgebras, we describe their Hom-Lie super structures, compute the cohomology spaces and characterize their infinitesimal deformations.

Valued Deformations of Algebras

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.

共变与变形理论的平均李代数

Cohomology and deformation theory of Averaging Leibniz algebras

本文研究了平均李代数的表示理论及上同调理论，并探讨了其微分和形式变形理论，揭示了上同调与变形上同调的关联。

$\mathfrak{sl}_2$的微分变形与扭曲的雅可比恒等式

Infinitesimal deformations of $\mathfrak{sl}_2$ with a twisted Jacobi identity

研究$\mathfrak{sl}_2$的微分变形，证明在满足特定条件时，变形后括号满足普通雅可比恒等式，解决Makhlouf和Silvestrov于2010年的猜想。

One-parameter formal deformations of Hom-Lie-Yamaguti algebras

This paper studies one-parameter formal deformations of Hom-Lie-Yamaguti algebras. The first, second and third cohomology groups on Hom-Lie-Yamaguti algebras extending ones on Lie-Yamaguti algebras are provided. It is proved that first and second cohomology groups are suitable to the deformation theory involving infinitesimals, equivalent deformations and rigidity. However, the third cohomology group is not suitable for the obstructions.

Cohomologies and deformations of pre-Lie-morphism triples

A (pre-)Lie-morphism triple consists of two (pre-)Lie algebras and a (pre-)Lie algebra homomorphism between them. We give chomologies of pre-Lie-morphism triples. As an application, we study the infinitesimal deformations of pre-Lie-morphism triples. Finally, we show that the cohomology of the pre-Lie-morphism triple can be deduced from a new cohomology of the Lie-morphism triple.

New Universal Deformation Formulas for deformation quantization

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…

Extensions of associative algebras

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.

Cohomology and deformations of O-operators on Hom-associative algebras

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.

Morphisms Cohomology and Deformations of Hom-algebras

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.

无穷小编织的量化与预Cartier双代数

Quantization of infinitesimal braidings and pre-Cartier quasi-bialgebras

本文研究了预Cartier双代数在量化无穷小编织中的作用，并探讨了相关概念如无穷小量子Yang-Baxter方程和Cartier环。

正特征中受限 Lie-Rinehart 代数的形变与上同调

Deformations and Cohomology of restricted Lie-Rinehart algebras in positive characteristic

本文研究正特征下受限 Lie-Rinehart 代数的形式形变，证明 p>2 时由 Evans-Fuchs 受限上同调控制，并为 p=2 构造新的上同调复形，同时分析了受限 Heisenberg 代数的结构与上同调。

通过形变到拟经典极限的泊松双代数

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。

结合Rota-Baxter代数的$L_\text{inf}$形变与同伦Rota-Baxter算子

The $L_\infty$-deformations of associative Rota-Baxter algebras and homotopy Rota-Baxter operators

该研究构造了控制相对Rota-Baxter代数的$L_\infty[1]$-代数，引入其对应的上同调研究无穷小形变，推导了余边界斜对称无穷小双代数的上同调，还定义了同伦相对Rota-Baxter算子并探究其与相关同伦代数的关系。

无穷小辫子结构与pre-Cartier双代数

Infinitesimal braidings and pre-Cartier bialgebras

提出辫子范畴的无穷小对应物及无穷小辫子结构，在(余)三角双代数(余)模范畴中详细阐述推广的pre-Cartier框架，刻画无穷小R-矩阵性质，并提供显式例子与重构定理。

受限李三系的上同调、扩张与形变

Cohomology, Extensions and Deformations of restricted Lie triple systems

本文构建了受限李三系的低阶上同调理论，定义了直到5阶的受限上链与上边界算子，并阐释了低阶上同调群在导子、模扩张、形式形变及系统扩张中的代数意义。

Global Deformations of the Witt Algebra of Krichever-Novikov Type

By considering non-trivial global deformations of the Witt (and the Virasoro) algebra given by geometric constructions it is shown that, despite their infinitesimal and formal rigidity, they are globally not rigid. This shows the need of a clear indication of the type of deformations considered. The families appearing are constructed as families of algebras of Krichever-Novikov type. They show up in a natural way in the global operator approach to the quantization of two-dimensional conformal field theory. In addition, a proof of the infinitesimal and formal rigidity of the Witt algebra is pre…

The evolution of the spectrum of a Frobenius Lie algebra under deformation

The category of Frobenius Lie algebras is stable under deformation, and here we examine explicit infinitesimal deformations of four and six dimensional Frobenius Lie algebras with the goal of understanding if the spectrum of a Frobenius Lie algebra can evolve under deformation. It can.

论 $C^{\infty}(\mathbb R^d)$ 的量子化

On the quantization of $C^{\infty}(\mathbb R^d)$

本文证明基本万有形变公式，刻画 $C^{\infty}(\mathbf R^d)$ 上初级障碍消失的无穷小形变与 Poisson 结构，并指出三维情形的特殊性质及其宇宙学启示。

与若尔当代数相关的紧对称空间的三明治算子与爱因斯坦形变

Sandwich operators and Einstein deformations of compact symmetric spaces related to Jordan algebras

本研究通过构造紧李代数的三明治算子、结合若尔当代数分析，解决了两类对称空间的爱因斯坦度量二阶形变问题，证明了多数可无穷小形变的不可约紧对称空间的非线性不稳定性。

Cohomology and Deformations of $n$-Lie algebra morphisms

The study of $n$-Lie algebras which are natural generalization of Lie algebras is motivated by Nambu Mechanics and recent developments in String Theory and M-branes. The purpose of this paper is to define cohomology complexes and study deformation theory of $n$-Lie algebra morphisms. We discuss infinitesimal deformations, equivalent deformations and obstructions. Moreover, we provide various examples.

On a class of n-Leibniz deformations of the simple Filippov algebras

We study the problem of the infinitesimal deformations of all real, simple, finite-dimensional Filippov (or n-Lie) algebras, considered as a class of n-Leibniz algebras characterized by having an n-bracket skewsymmetric in its n-1 first arguments. We prove that all n>3 simple finite-dimensional Filippov algebras are rigid as n-Leibniz algebras of this class. This rigidity also holds for the Leibniz deformations of the semisimple n=2 Filippov (i.e., Lie) algebras. The n=3 simple FAs, however, admit a non-trivial one-parameter infinitesimal 3-Leibniz algebra deformation. We also show that the $n…