Meighan I. Dillon
The Z-grading determined by a long simple root of an affine or finite type Lie algebra arises from an adjoint or cominuscule representation of a lower rank semi-simple complex Lie algebra. Analysis of the relationship between the grading and the representation leads to an extension of Kac's construction of nontwisted affine Lie algebras.
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
For any finitely generated abelian group $Q$, we reduce the problem of classification of $Q$-graded simple Lie algebras over an algebraically closed field of "good" characteristic to the problem of classification of gradings on simple Lie algebras. In particular, we obtain the full classification of finite-dimensional $Q$-graded simple Lie algebras over any algebraically closed field of characteristic $0$ based on the recent classification of gradings on finite dimensional simple Lie algebras. We also reduce classification of simple graded modules over any $Q$-graded Lie algebra (not necessari…
We develop general results on centroids of Lie algebras and apply them to determine the centroid of extended affine Lie algebras, loop-like and Kac-Moody Lie algebras, and Lie algebras graded by finite root systems.
The filiform and the quasi-filiform Lie algebras form a special class of nilpotent Lie algebras. The aim of this paper is to compute the index and provide regular vectors of this two class of nilpotent Lie algebras. we consider the graded filiform Lie algebras $L_{n}$, $Q_{n}$, the $n$-dimensional filiform Lie algebras for $n$ $<8$, also the graded quasi-filiform Lie algebras and finally Lie algebras whose nilradical is $Q_{2n}$.
We determine the Lie superalgebras over fields of characteristic zero that are graded by the root system A(n,n) of the special linear Lie superalgebra psl(n+1,n+1).
We compute the cohomology with trivial coefficients of two graded infinite-dimensional Lie algebras of maximal class, give explicit formulas for their representative cocycles. Also we discuss the relations with combinatorics and representation theory.
Let g_A (respectively, g_A(μ)) be the graded multi-loop Lie algebra (respectively graded twisted multi-loop Lie algebra)" associated with the simple finite dimensional Lie algebra g over the complex field C. In this paper, we prove that irreducible integrable g_A(μ)-modules with finite dimensional weight spaces are either highest weight modules or their duals and classify the isomorphism classes of irreducible integrable g_A-modules and g_A(μ)-modules with finite dimensional weight spaces.
This paper is primarily concerned with generalized reduced Verma modules over $\mathbb{Z}$-graded modular Lie superalgebras. Some properties of the generalized reduced Verma modules and the coinduced modules are obtained. Moreover, the invariant forms on the generalized reduced Verma modules are considered. In particular, we prove that the generalized reduced Verma module is isomorphic to the mixed product for modules of $\mathbb{Z}$-graded modular Lie superalgebras of Cartan type.
We classify two classes of B_2-graded Lie algebras which have a second compatible grading by an abelian group A: (a) graded-simple Lie algebras for A torsion-free and (b) division-A-graded Lie algebras. Our results describe the centreless cores of a class of affine reflection Lie algebras, hence apply in particular to the centreless cores of extended affine Lie algebras, the so-called Lie tori, for which we recover results of Allison-Gao and Faulkner. Our classification (b) extends a recent result of Benkart-Yoshii. Both classifications are consequences of a new description of Jordan algebras …
We compute explicitly the adjoint cohomology of two N-graded Lie algebras of maximal class (infinite dimensional filiform Lie algebras) m_0 and m_2. It is known that up to an isomorphism there are only three N-graded Lie algebras of the maximal class. The third algebra from this list is the "positive" part L_1 of the Witt (or Virasoro) algebra and its adjoint cohomology was computed earlier by Feigin and Fukhs. We show that the total space H*(m_j,m_j) is "almost" isomorphic to the completed tensor product of the algebra m_j by scalar cohomology space H^*(m_j), j=0,2.
Cup product, Frölicher-Nijenhuis bracket and the derived bracket associated to Hom-Lie algebras
本文围绕Hom-李代数构造了杯积括号、Frölicher-Nijenhuis括号和导出括号等新分次李代数结构，并分别应用于态射形变、Nijenhuis算子及Rota-Baxter算子研究。
Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The invariants appearing in this classification are computed in the case when L is simple classical (except for type D4, where a partial result is given). In particular, we obtain criteria to determine when a finite-dimensional simple L-module admits a G-grading making it a graded L-module.
Compatible root graded anti-pre-Lie algebraic structures on finite-dimensional complex simple Lie algebras
本研究借助${\rm sl_2(\C)}$表示理论探究有限维复单李代数的相容根阶化反预李代数结构，证明仅${\rm sl_2(\C)}$上存在且仅存在一个此类结构，其余有限维复单李代数均不存在该结构。
We use a fermionic extension of the bosonic module to obtain a class of B(0,N)-graded Lie superalgebras with nontrivial central extensions.
We give new applications of graded Lie algebras to: identities of standard polynomials, deformation theory of quadratic Lie algebras, cyclic cohomology of quadratic Lie algebras, $2k$-Lie algebras, generalized Poisson brackets and so on.
In this master's thesis, we recall the definitions and basic results for Lie superalgebras. We specify the definition for Klein graded Lie algebras and, motivated by well known results for Lie superalgebras, we prove similar results for Klein graded Lie algebras. More precisely, we state the theorems of Poincare-Birkhoff-Witt and Ado's, as well as Schur's lemma. Moreover, we present two examples of Klein graded algebra.
We give a block decomposition of the equivariant derived category arising from a cyclically graded Lie algebra. This generalizes certain aspects of the generalized Springer correspondence to the graded setting.
Cuspidal character sheaves on graded exceptional Lie algebras: stable gradings
本文研究了所有（GIT）稳定分级例外李代数的尖锐特征sheaves的显式确定方法，揭示了其核心贡献和数学结构。
We consider a fixed block for the equivariant perverse sheaves with nilpotent support in the $1$-graded ccomponent of a semisimple cyclically graded Lie algebra. We give a combinatorial parametrization of the simple objects in that block.
The aim of this paper is to study relations between regular reductive PVs with one-dimensional scalar multiplication and the structure of graded Lie algebras. We will show that the regularity of such PVs is described by an $\mathfrak{sl}_2$-triplet of a graded Lie algebra.
In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalgebras.
Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras
本文在复数域上，对标准分次下所有简单有限维向量李超代数的极大分次可解子代数进行了完整分类。
An important problem in the representation theory of affine and toroidal Lie algebras is to classify all possible irreducible integrable modules with finite dimensional weight spaces. Recently the irreducible integrable modules having finite dimensional weight spaces with non-trivial central action have been classified for a more general class of Lie algebras, namely the graded Lie tori. In this paper, we classify all the irreducible integrable modules with finite dimensional weight spaces for this graded Lie tori where the central elements act trivially. Thus we ultimately obtain all the simp…
We define a pair of symplectic Dirac operators $(D^+,D^-)$ in an algebraic setting motivated by the analogy with the algebraic orthogonal Dirac operators in representation theory. We work in the settings of $\mathbb Z/2$-graded quadratic Lie algebras $\mathfrak g=\mathfrak k+\mathfrak p$ and of graded affine Hecke algebras $\mathbb H$.
In a previous paper by the authors, we obtain the first example of a finitely freely generated simple $\mathbb Z$-graded Lie conformal algebra of linear growth that cannot be embedded into any general Lie conformal algebra. In this paper, we obtain, as a byproduct, another class of such Lie conformal algebras by classifying $\mathbb Z$-graded simple Lie conformal algebras ${\cal G}=\oplus_{i=-1}^\infty{\cal G}_i$ satisfying the following, (1) ${\cal G}_0\cong{\rm Vir}$, the Virasoro conformal algebra; (2) Each ${\cal G}_i$ for $i\ge-1$ is a ${\rm Vir}$-module of rank one. These algebras includ…
Character sheaves for classical graded Lie algebras
本文针对源自特殊线性群内自同构的分次李代数与Vinberg II型典型分次李代数，开展其特征层的研究工作。
Any graded restricted simple Lie algebra of Cartan type contains a subalgebra isomorphic to the Witt algebra over a field of prime characteristic. As some analogue of study on branching rules for restricted non-classical Lie algebras, it is shown that each graded restricted simple Lie algebra of Cartan type can be decomposed into a direct sum of restricted baby Verma modules and simple modules as an adjoint module over the Witt algebra. In particular, the composition factors are precisely determined.
In this paper we construct representations of certain graded double affine Hecke algebras (DAHA) with possibly unequal parameters from geometry. More precisely, starting with a simple Lie algebra $\mathfrak{g}$ together with a $\mathbb{Z}/m\mathbb{Z}$-grading $\oplus_{i}\mathfrak{g}_{i}$ and a block of $G_{\underline{0}}$-equivariant complexes on the nilpotent cone of $\mathfrak{g}_{\underline{1}}$ as introduced in \cite{LY1}, we attach a graded DAHA and construct its action on the direct sum of spiral inductions in that block. This generalizes results of Vasserot \cite{V} and Oblomkov-Yun \ci…
In this paper, we study Whittaker modules for graded Lie algebras. We define Whittaker modules for a class of graded Lie algebras and obtain a bijective correspondence between the set of isomorphism classes of Whittaker modules and the set of ideals of a polynomial ring, parallel to a result from the classical setting and the case of the Virasoro algebra. As a consequence of this, we obtain a classification of simple Whittaker modules for such algebras. Also, we study some concrete algebras as examples.
Cuspidal character sheaves on graded Lie algebras II
本文研究了有级数李代数上尖点特征sheaves的分类及其支持，为明确分类这些sheaves提供了重要进展。
We use fermionic representations to obtain a class of BC$_{{}_{\text N}}$-graded Lie algebras coordinatized by quantum tori with nontrivial central extensions.
We show that the Temperley-Lieb algebra of type $A$ and the blob algebra (also known as the Temperley-Lieb algebra of type $ B$) at roots of unity are $ \mathbb Z$-graded algebras.We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.
Let g=g_0+ g_1 be a simple Z_2-graded Lie algebra and let b_0 be a fixed Borel subalgebra of g_0. We describe and enumerate the abelian b_0-stable subalgebras of g_1.
In this paper, we shall give a way to construct a graded Lie algebra $L(\mathfrak{g},ρ,V,{\cal V},B_0)$ from a standard pentad $(\mathfrak{g},ρ,V,{\cal V},B_0)$ which consists of a Lie algebra $\mathfrak{g}$ which has a non-degenerate invariant bilinear form $B_0$ and $\mathfrak{g}$-modules $(ρ, V)$ and ${\cal V}\subset \mathrm {Hom }(V,\mathfrak{k})$ all defined over a field $\mathfrak{k}$. In general, we do not assume that these objects are finite-dimensional. We can embed the objects $\mathfrak{g},ρ,V,{\cal V}$ into $L(\mathfrak{g},ρ,V,{\cal V},B_0)$. Moreover, we construct specific positiv…
Invariant systems and character sheaves for graded Lie algebras
该研究通过引入微分方程不变系统构造分次李代数上的特征层，并与既有结果共同给出Vinberg I型经典情形的全部尖点特征层。
Let $\mathbb{k}$ be an algebraically closed field of characteristic $ p>0. $ In this short note, we illustrate a class of Lie superalgebras over $ \mathbb{k} $ such that the category of restricted supermodules is of one block. As an application, if $ p>3 $ and $ \mathfrak{g} $ is a graded restricted Cartan type Lie superalgebra of type W, S and H, then the category of restricted $ \mathfrak{g} $ supermodules is of one block.
In this note we propose a method to classify homogeneous nilpotent elements in a real $Z_m$-graded semisimple Lie algebra $g$. Using this we describe the set of orbits of homogeneous elements in a real $Z_2$-graded semisimple Lie algebra. A classification of 4-vectors (resp. 4-forms) on $R^8$ can be given using this method.
In this paper, we study certain ad-nilpotent subalgebras contained in the non-zero graded portion of a simple Z_n-graded Lie algebra. These subalgebras respect the grading on the Lie algebra and are modules for a Borel subalgebra for the grade-zero Lie subalgebra. We show that semisimple elements in such subalgebras lie in the center of the subalgebra, and we provide a classification of these subalgebras whose weight space decompositions have only non-zero weights.
Representations of non-finitely graded Lie algebras related to Virasoro algebra
本文研究与Virasoro代数相关的非有限分次李代数 $\mathcal{W}(ε)$ 的表示，完全分类了其秩为1的自由 $\mathcal{U}(\mathfrak h)$-模，确定了这些模的单性与同构类，并揭示了其与其他分次李代数模结构的显著差异。
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。
Cuspidal character sheaves on graded Lie algebras
本文通过邻近循环构造和傅里叶-萨托变换，完成了文金类型I分层经典李代数极小特征特征sheaves的分类。
For a semisimple Lie algebra over the complex numbers, Dynkin (1952) developed an algorithm to classify the regular semisimple subalgebras, up to conjugacy by the inner automorphism group. For a graded semisimple Lie algebra over the complex numbers, Vinberg (1979) showed that a classification of a certain type of regular subalgebras (called carrier algebras) yields a classification of the nilpotent orbits in a homogeneous component of that Lie algebra. Here we consider these problems for (graded) semisimple Lie algebras over the real numbers. First, we describe an algorithm to classify the re…
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.
In this paper, we introduce the notion of completely non-trivial module of a Lie conformal algebra. By this notion, we classify all finite irreducible modules of a class of $\mathbb{Z}^+$-graded Lie conformal algebras $\mathcal{L}=\bigoplus_{i=0}^{\infty} \mathbb{C}[\partial]L_i$ satisfying $ [{L_0}_λL_0]=(\partial+2λ)L_0,$ and $[{L_1}_λL_i]\neq 0$ for any $i\in \mathbb{Z}^+$. These Lie conformal algebras include Block type Lie conformal algebra $\mathcal{B}(p)$ and map Virasoro Lie conformal algebra $\mathcal{V}(\mathbb{C}[T])=Vir\otimes \mathbb{C}[T]$. As a result, we show that all non-trivi…
Let L be the Lie algebra of Block type over a field F of characteristic zero, defined with basis {L_{i,j} | i,j\in Z} and relations [L_{i,j},L_{k,l}]=((j+1)k-(l+1)i)L_{i+k,j+l}. Then L is Z^2-graded. In this paper, Z^2-graded L-modules of the intermediate series are classified.
We provide a new proof of the super duality equivalence between infinite-rank parabolic BGG categories of general linear Lie (super) algebras conjectured by Cheng and Wang and first proved by Cheng and Lam. We do this by establishing a new uniqueness theorem for tensor product categorifications motivated by work of Brundan, Losev, and Webster. Moreover we show that these BGG categories have Koszul graded lifts and super duality can be lifted to a graded equivalence.
In this paper we investigate Besov spaces on graded Lie groups. We prove a Nikolskii type inequality (or the Reverse Hölder inequality) on graded Lie groups and as consequence we obtain embeddings of Besov spaces. We prove a version of the Littlewood-Paley theorem on graded Lie groups. The results are applied to obtain embedding properties of Besov spaces and multiplier theorems for both spectral and Fourier multipliers in Besov spaces on graded Lie groups. In particular, we give a number of sufficient conditions for the boundedness of Fourier multipliers in Besov spaces.
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。
In this paper we study multipliers on graded nilpotent Lie groups defined via group Fourier transform. More precisely, we show that Hörmander type conditions on the Fourier multipliers imply $L^p$-boundedness. We express these conditions using difference operators and positive Rockland operators. We also obtain a more refined condition using Sobolev spaces on the dual of the group which are defined and studied in this paper.
Reduction theory for stably graded Lie algebras
本文为稳定分次李代数的Vinberg表示定义约化协变，并给出奇分裂特殊正交群的LLL算法类比，用于约化与奇超椭圆曲线2-下降相关的向量系数。
In this note we present a symbolic pseudo-differential calculus on graded nilpotent Lie groups and, as an application, a version of the sharp Garding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.
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。
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。
Character Sheaves for Graded Lie Algebras: Stable Gradings
本研究针对稳定分次李代数，通过分析[GVX2]中邻近周期层的傅里叶变换，构造出推测为尖点特征层的满支撑特征层，涉及单位根处复反射群相关Hecke代数的不可约表示。
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。
In this paper, we present first results of our investigation regarding symbolic pseudo-differential calculi on nilpotent Lie groups. On any graded Lie group, we define classes of symbols using difference operators. The operators are obtained from these symbols via the natural quantisation given by the representation theory. They form an algebra of operators which shares many properties with the usual Hormander calculus.
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.
In this note, we consider a non-commutative analogy of the classical Fefferman multiplier problem for the ball. More precisely, if $χ$ is the characteristic function of the unit interval $I=[0,1],$ we investigate a family of differential operators $\mathcal{R}$ on a graded Lie group $G,$ for which the multipliers $χ(\mathcal{R})$ are bounded on $L^p(G),$ if and only if $p=2.$
Estimates for the full maximal function on graded Lie groups
本文将 Bourgain 关于欧氏空间球面平均完全极大函数的限制弱型不等式推广到分级李群上，借助群傅里叶变换和正 Rockland 算子的符号建立相应估计。
In this work we extend the $L^1$-Björk-Sjölin theory of strongly singular convolution operators to arbitrary graded Lie groups. Our criteria are presented in terms of the oscillating Hörmander condition due to Björk and Sjölin of the kernel of the operator, and the decay of its group Fourier transform is measured in terms of the infinitesimal representation of an arbitrary Rockland operator. The historical result by Björk and Sjölin is re-obtained in the case of the Euclidean space.
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。