On Z-graded loop Lie algebras, loop groups, and Toda equations
Kh. S. Nirov, A. V. Razumov
Toda方程与扭环群被认为是。这类方程由相应的扭圈李代数的Z-阶化来指定。讨论了与具有可积Z-阶的扭圈李代数相关的Toda方程的分类。
Toda equations associated with twisted loop groups are considered. Such equations are specified by Z-gradations of the corresponding twisted loop Lie algebras. The classification of Toda equations related to twisted loop Lie algebras with integrable Z-gradations is discussed.
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N=(1|1) supersymmetric dispersionless Toda lattice hierarchy
推广了超代数中的分次交换子，提出了一种新的具有对合的分次算子空间上的括号运算。我们研究了这类运算的性质，证明了二维N=（1）的Lax表示|1）超对称Toda格族可以通过广义括号运算实现;这对于构造该格族的半经典（连续）极限是重要的。我们构造了N=（1）的连续极限|1）Toda格系，无色散N=（1| 1）户田阶层。在此极限下，我们得到了Lax表示，广义分次括号成为分次相超空间上相应的泊松括号。我们发现无色散N=（1| 1）超对称Toda方程。
Fermionic one- and two-dimensional Toda lattice hierarchies and their bi-Hamiltonian structures
通过给出相应的拉克斯对表示，我们提出了一个广泛的可积二维费米Toda格族，它包括二维N=（2| 2）和N=（0| 2）作为特例的超对称TL族。通过施加适当的约束，将其归约到一维情况，我们推导出相应的1D费米子TL体系。我们利用超代数中分次交换子的对合推广的分次算子空间上的广义分次括号，发展了广义分次R-矩阵形式，它允许人们在Hamilton形式的框架下描述这些层次，并构造它们的前两个Hamilton结构.第一个哈密顿结构得到的玻色子和费米子Lax运营商，而第二个哈密顿结构建立玻色子Lax运营商。提出了算子形式的分次修正Yang-Baxter方程，并证明了对于分次反对称R-矩阵类，它等价于分次经典Yang-Baxter方程的张量形式.
Hamiltonian structures of fermionic two-dimensional Toda lattice hierarchies
通过给出相应的拉克斯对表示，我们提出了一个广泛的可积二维费米Toda格族，它包括二维N=（2| 2）和N=（0| 2）作为特例的超对称TL族。利用对合分次算子空间上的广义分次括号，推广了超代数中的分次交换子，建立了广义分次R-矩阵形式，使人们能够在Hamilton形式的框架下描述这些层次，并构造它们的前两个Hamilton结构.第一个哈密顿结构得到的玻色子和费米子Lax运营商，而第二个哈密顿结构建立玻色子Lax运营商。
We consider discrete one dimensional nonlinear equations and present the procedure of lifting them to Z-graded graphs. We identify conditions which allow one to lift one dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons {for static potentials} on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z-graded graph.
Toda equations associated with loop groups of complex classical Lie groups
Toda方程由李群的选择和它的李代数的$\mathbb Z$ -阶化来指定。给出了复经典李群的圈群所对应的Toda方程的显式描述，其中复经典李群的李代数被赋予可积的$\mathbb Z$ -阶化，其阶化子空间为有限维.
Graded Lie Algebra Generating of Parastatistical Algebraic Structure
引入了一种新的分次李代数（我们称之为$Z_{2,2}$分次李代数）作为仿对称理论的框架。通过选择合适的$Z_{2,2}$分次李代数的玻色子空间，利用相应的广义Jacobi恒等式，得到了仿统计量的完整代数结构。
N=5 three-algebras and 5-graded Lie superalgebras
本文讨论了N=6三代数到N=5三代数的推广，并将其推广到反李三系和II型基本李超代数。然后，我们表明，在反李三系中定义的结构常数同意在三维N=5超共形理论。
This paper presents the classification, over the fields of real and complex numbers, of the minimal ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie algebras and Lie superalgebras spanned by $4$ generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional $…
`Mixed' Jordan-Lie Superalgebra
一个代数A没有遇到任何通常的代数簇或超簇的介绍。A是四元数的分次变形形式，其结构类似于大久保和神谷定义的Jordan-李超代数，但它既不是纯结合李超代数，也不是纯反结合Jordan-李超代数。相反，它表现出一种新的结合性，这里称之为“有序分次结合性”，它在某种程度上介于纯结合性和纯反结合性之间。除了分次结合性之外，A的生成元还服从在通常的李超代数和分次Jordan-李代数中遇到的分次对易关系。它们还满足新的分次Jacobi恒等式，结合了未分次李，分次李和分次Jordan-李代数的生成元所服从的Jacobi的特征。主要由于这三个特征，A被称为“混合”Jordan-李超代数。本文定义了A，并将它与Okubo和Kamiya定义的Jordan-Lie超代数进行了比较。
A New Kind of Graded Lie Algebra and Parastatistical Supersymmetry
本文将通常的$Z_2$分次李代数推广到一种新的形式，称之为$Z_{2,2}$分次李代数.结果表明，$Z_{2,2}$分次李代数与准统计量之间存在着密切的联系，因此，$Z_{2,2}$可用于研究和分析准粒子系统的各种对称性和超对称性.
Three Examples of Graded Lie Groups
本文构造了三类分次李群，给出其两种视角的构建方式并证明其李代数的同构性质，为分次李群研究提供了具体实例基础。
Lie Superalgebras Graded by the Root System $A(m,n)$
我们确定了由A型基本经典单李超代数的根系分次的李超代数$(m,n)$。
Differential operators on Lie and graded Lie algebras
结合代数上的微分算子理论没有直接推广到非结合代数上。我们考虑李代数上的微分算子。一个关键点是李代数中的乘法是它的导子。李代数上的高阶微分算子被定义为一阶微分算子的合成。定义了李代数上的Chevalley-Eilenberg微分.有限维李代数，泊松代数，代数的向量场，代数的典型交换关系的例子被认为是。分次李代数上的微分算子的定义与李代数上的微分算子相同。
We establish a Gelfand-Naimark-Segal construction which yields a correspondence between cyclic unitary representations and positive definite superfunctions of a general class of $\mathbb Z_2^n$-graded Lie supergroups.
$\mathbf{{\mathbb Z}_2\ \times {\mathbb Z}_2}$-graded Lie (super)algebras and generalized quantum statistics
本文在$\mathbb{Z}_2\times\mathbb{Z}_2$-分次李（超）代数框架下，用根向量刻画顺统计算子，并构造了满足相对顺统计关系的多系综系统。
Generic graded contractions of Lie algebras
本文研究了李代数的通用分层收缩，通过群上同调、仿射代数几何和单向范畴的方法，分类了具有固定支撑的收缩，并建立了关于等价猜想的函子版本。
Quantum mechanical systems whose symmetry is given by $\mathbb{Z}_2^3$-graded version of superconformal algebra are introduced. This is done by finding a realization of a $\mathbb{Z}_2^3$-graded Lie superalgebra in terms of a standard Lie superalgebra and the Clifford algebra. The realization allows us to map many models of superconformal quantum mechanics (SCQM) to their $\mathbb{Z}_2^3$-graded extensions. It is observed that for the simplest SCQM with $\mathfrak{osp}(1|2)$ symmetry there exist two inequivalent $\mathbb{Z}_2^3$-graded extensions. Applying the standard prescription of conforma…
Finite-dimensional $\mathbb{Z}$-graded Lie algebras
本文研究了有限维 $\mathbb{Z}$-分次 Lie 代数的结构与表示理论，扩展了有限维半单 Lie 代数理论，并揭示了 Heisenberg 与共形 Galilei 代数等物理实例的分次结构应用潜力。
Graded Casimir Elements and Central Extensions of Color Lie Algebras
本文研究了色李代数及其环代数的分级Casimir元素和中心扩张的构造方法，并通过三个例子展示了此类结构的存在性。
Graded Lie superalgebras from embedding tensors
本文通过嵌入张量构造分层李超代数，并揭示其与李代数模之间的关系。
Controlling graded Lie algebras and cohomologies of double Lie algebras
本文构造控制双李代数结构的分次李代数，并证明其扭转后与Fairon-Valeri复形同构，进而将双李代数上同调等同于矩阵代数上斜对称Rota-Baxter算子派生结合代数的循环上同调。
We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numerous natural examples. Note that $\mathcal{VB}$-groupoids, extensively studied in the recent literature, form just the particular case of weighted Lie groupoids of degree one. We examine the Lie theory related to weighted groupoids and weighted Lie algebroids, objects defined in a …
Solvable Lie algebras with naturally graded nilradicals and their invariants
研究了具有最大幂零指标的分次幂零根和余维为1的Heisenberg子代数的不可分解可解李代数，并计算了它们的广义Casimir不变量。证明了秩一可解代数具有接触形式，这意味着存在一个相关联的动力系统。此外，由于幂零根的二次Casimir算子的结构，这些代数包含一个维数为$2n-1$的极大非交换准经典李代数，表明规范理论（带鬼）在这些子代数上是可能的。
Categorification of Pre-Lie Algebras and Solutions of 2-graded Classical Yang-Baxter Equations
在本文中，我们引入了预李2-代数的概念，它是预李代数的分类。证明了2-项pre-Lie $_\infty$ -代数范畴与2-项pre-Lie 2-代数范畴是等价的。利用准李代数的第三上同调对骨架准李2-代数进行了分类。证明了准李代数的交叉模与严格准李2-代数一一对应。在Lie 2-代数上引入了$\mathcal O$ -算子，它可以用来构造准Lie 2-代数.作为应用，我们给出了一类半直积李2-代数上的2-阶化经典Yang-Baxter方程的解。
We generalize the Donagi and Witten construction of a first obstruction class for splitting of a supermanifold via differential operators using the theory of $n$-fold vector bundles and graded manifolds. Applying the generalized Donagi--Witten construction we obtain a family of embeddings of the category of supermanifolds into the category of $n$-fold vector bundles and into the category of graded manifolds. This leads to a realization of any non-split supermanifold in terms of a collection of vector bundles and some morphism between them. Further we study the images of these embeddings into t…
From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets
设g是长度为k的Moore复形为Ng的单李代数.设G是对g积分的单李群，它在每个单水平上单连通.我们利用G的分类空间的1-jet，从g出发，构造了一个k-李代数L。这样构造的李k-代数L实际上是一个微分分次李代数。微分和括号明确地描述在Ng的相应的k-超交叉复结构的条款（的一部分）。这个结果可以看作是Quillen关于单纯李代数和dg-李代数之间的连接的（纯代数的）构造的一个几何解释。
In this paper we present a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups. Moreover, using the comparison principle we obtain blow-up type results and global in $t$-boundedness of solutions of nonlinear equations for the heat $p$-sub-Laplacian on stratified Lie groups. In particular, this paper generalises and extends previous results obtained by the first author and Suragan in [RS18].
In this paper we study the Cauchy problem for diffusion equations associated to a class of strongly hypoelliptic pseudo-differential operators on graded Lie groups. To do so, we develop a global complex functional calculus on graded Lie groups in order to analyse the corresponding energy estimates. One of the main aspects of this complex functional calculus is that for the $(ρ,δ)$-Euclidean Hörmander classes we recover the standard functional calculus developed by Seeley [38]. In consequence the Gårding inequality that we prove for arbitrary graded Lie groups absorbs the historical 1953's ineq…
We present a novel realisation of the $\mathbb{Z}_2\times\mathbb{Z}_2$-graded Lie superalgebra $\mathfrak{gl}(m_1,m_2|n_1,n_2)$ inside an algebraic extension of the enveloping algebra of the $\mathbb{Z}_2$-graded Lie superalgebra $\mathfrak{gl}(m|n)$, with $m=m_1+m_2$ and $n=n_1+n_2$. A consequence of this realisation is that the representations of $\mathfrak{gl}(m|n)$ "lift up" to representations of $\mathfrak{gl}(m_1,m_2|n_1,n_2)$, with matrix elements differing only by a sign, which we are able to characterise concisely.
Graded anti-pre-Lie algebraic structures on Witt and Virasoro algebras
本文通过分类Witt代数的不可分解权表示，给出了其上的分次反pre-Lie代数结构及同构分类，并证明了Virasoro代数上不存在满足自然条件的此类结构。
This thesis studies the representation theory and linear structures of $\mathcal{Q}$-manifolds and higher Lie algebroids. We introduce differential graded modules (or for short DG-modules) of $\mathcal{Q}$-manifolds and the equivalent notion of representations up to homotopy in the case of Lie $n$-algebroids ($n\in\mathbb{N}$), as generalisations of the homonymous structures that exist already in the case of ordinary Lie algebroids. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy of Lie $n$-algebr…
Matrix structure of classical ${\mathbb Z}_2 \times {\mathbb Z}_2$ graded Lie algebras
本文研究了 ${\mathbb Z}_2\times{\mathbb Z}_2$ 分次李代数的矩阵结构，构造了对应于经典李代数的新类，并探讨了其在副统计学中的可能应用。
On classical $Z_2 \times Z_2$-graded Lie algebras
本文基于定义矩阵构造了对应于经典 Lie 代数的各类 $Z_2 \ imes Z_2$-分次 Lie 代数，其中 $B$、$C$ 和 $D$ 型的构造为全新结果，并探讨了其在 parastatistics 中的可能应用。
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