Jean-Yves Charbonnel
In this article, we are interested in some subspaces of the exterior algebra of a simple Lie algebra g. In particular, we prove that some graded subspaces of degree d generate the g-module d (g) for some integers d.
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On special covariants in the exterior algebra of a simple Lie algebra
研究了由小伴随表示的拷贝线性张成的单复李代数的外代数的子空间，或者在无迹矩阵的李代数的情况下，由定义表示的n次对称幂的拷贝线性张成的单复李代数的外代数的子空间.作为主要结果，我们证明了这个子空间是由除最高次本原不变量以外的所有本原不变量生成的外代数的子代数上的自由模。
The adjoint representation inside the exterior algebra of a simple Lie algebra
对于一个简单的复李代数$\mathfrak g$，我们研究了不变量空间$A=\left( \bigwedge \mathfrak g^*\otimes\mathfrak g^*\right)^{\mathfrak g}$（它描述了$ \bigwedge \mathfrak g^*$中$\mathfrak g$型的同型分量）作为不变量代数$\left(\bigwedge \mathfrak g^*\right)^{\mathfrak g}$上的模。作为主要结果，我们证明了$A$是$(\bigwedge \mathfrak g^*)^{\mathfrak g}$中除最高次本原不变量外的所有本原不变量所生成的外代数上秩为$\mathfrak g$秩两倍的自由模.
As a first application of the double affine Hecke algebra with unequal parameters on Weyl orbits to representation theory of semisimple Lie algebras, we find the graded multiplicities of the trivial module and of the little adjoint module in the exterior algebra of the little adjoint module of a simple Lie algebra $\,\mathfrak{g}\,$ with a non-simply laced Dynkin diagram. We prove that in type $\,B, C\,$ or $\,F\,$ these multiplicities can be expressed in terms of special exponents of positive long roots in the dual root system of $\,\mathfrak{g}.\,$
Exterior powers of a parabolic Springer sheaf on a Lie algebra
本文计算了一般线性李代数上对应于(1, n-1)型极大抛物子群的抛物Springer层在加法卷积下的外幂，并证明其同构于对称群置换表示外幂对应的半单反常层。
Casimir operators, abelian subspaces and u-cohomology
这篇综述论文阐述了Kostant等人关于简单李代数的外代数与Casimir算子在其上的作用之间关系的新旧结果。我们的阐述依赖于u-上同调，并且基本上是自包含的。
Let $\mathfrak{g}$ be a real finite-dimensional Lie algebra containing pointed generating invariant closed convex cones. We determine those derivations $D$ of $\mathfrak{g}$ which induce a 3-grading of the form $\mathfrak{g} = \mathfrak{g}_{-1} \oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1$ on $\mathfrak{g}$ such that the $(\pm 1)$-eigenspaces $\mathfrak{g}_{\pm 1}$ of $D$ are generated by the intersections with generating cones of the form $W = \mathcal{O}_f^*$, where $\mathcal{O}_f$ is the coadjoint orbit of a linear functional $f \in \mathfrak{z}(\mathfrak{g})^*$ and $\mathcal{O}_f^*$ is the …
Let $U$ be a finite dimentional vector space over $\mathbb R$ or $\mathbb C$, and let $ρ:G\to GL(U)$ be a representation of a connected Lie group $G$. A linear subspace $V\subset U$ is called universal if every orbit of $G$ meets $V$. We study universal subspaces for Lie groups, especially compact Lie groups. Jinpeng and Doković approached universality for compact groups through a certain topological obstruction. They showed that the non-vanishing of the obstruction class is sufficient for the universality of $V$, and asked whether it is also necessary under certain conditions. In this article…
Exterior powers of the reflection representation in the cohomology of Springer fibres
设$H^*(\calB_e)$为单李代数$\g$中幂零元$e$的Springer纤维的上同调，Weyl群$W$作用于该单李代数上.令$Λ^i V$表示$W$的反射表示的$i$次外幂。在假定$e$在Levi子代数中是正则的并且满足一定的附加条件的情况下，我们确定了$Λ^i V$在分次表示$H^*(\calB_e)$中出现的次数，如果$\g$是A、B或C型的，则该附加条件自动成立.这部分地证实了Lehrer-Shoji的一个猜想，推广了Solomon在$e=0$情形下和Lehrer-Shoji在$i=1$情形下的结果.证明了$(H^*(\calB_e) \otimes Λ^* V)^W$是它的子空间$(H^*(\calB_e)\otimes V)^W$上的自由外代数。
When we consider the action of a finite group on a polynomial ring, a polynomial unchanged by the action is called an invariant polynomial. A famous result of Noether states that in characteristic zero the maximal degree of a minimal invariant polynomial is bounded above by the order of the group. Our work establishes that the same bound holds for invariant skew polynomials in the exterior algebra. Our approach to the problem relies on a theorem of Derksen that connects invariant theory to the study of ideals of subspace arrangements. We adapt his proof over the polynomial ring to the exterior…
A polynomial ring with rational coefficients is an irreducible representation of Lie algebras of endomorphisms of exterior powers of a infinite countable dimensional $\mathbb{Q}$-vector space. We give an explicit description of it, using suitable vertex operators on exterior algebras, which mimick those occurring in the bosonic vertex representation of the Lie algebra $gl_\infty$, due to Date--Jimbo--Kashiwara and Miwa (DJKM).
Universal subspaces for compact Lie groups
对于连通紧李群G在有限维实向量空间U和U的一个在G的极大环面下不变的子空间V中的表示，我们得到了V满足U中所有G-轨道的一个充分条件，这在某些情况下也是必要的.证明利用了旗流形的上同调和外尔群的不变量理论。然后，我们将我们的条件应用于U（n），Sp（n）和SO（n）在C，H和R上的$n\times n$矩阵空间中的共轭表示。特别是，我们得到一个有趣的推广舒尔的三角化定理。
Graded multiplicities in the exterior algebra
我们知道一个半单李代数的伴随表示在它自己的外代数中的重数，但是它的副本如何在它的外幂之间分布呢？答案（分次多重性）是借助麦克唐纳多项式得到的。
Crowned Lie groups and nets of real subspaces
本文为连通李群引入复加冕域，利用反酉表示解析延拓构造满足代数量子场论条件的实子空间网，并以欧拉元素正则条件刻画其存在性。
The exterior algebra and `Spin' of an orthogonal g-module
Kostant的一个著名结果给出了李代数$\frak g$的外代数的G-模结构的描述。我们将这个结果推广到对称空间的各向同性表示。如果$\frak g={\frak g}_0+{\frak g_1}$是一个单李代数的Z_2-分次，我们明确地描述了一个${\frak g}_0$ -模$Spin_0({\frak g}_1)$，使得${\frak g}_1$的外代数是这个模的张量平方乘以2的某个幂.虽然$Spin_0({\frak g}_1)$通常是可约的，但我们证明了${\frak g}_0$的Casimir元总是标量化地作用在它上面，并给出了具有斜不变量外代数的单代数群的所有正交表示的分类.
Quasi-bigèbres de Lie et cohomologie d'algèbre de Lie
李拟双代数是由Drinfeld引入的李双代数的自然推广。对于任意有限维（G，μ，γ，μ？）的李拟双代数结构，对应于D = G \oplus G* 上的一个李代数结构，称为给定李拟双代数的二重。证明了在G的外代数ΛG上存在D-模结构，并建立了ΛD与End（ΛG）之间的D-模同构，D通过伴随作用于ΛD.
We obtain a presentation of principal subspaces of basic modules for the twisted affine Kac-Moody Lie algebras of type $A_{2n-1}^{(2)}$, $D_n^{(2)}$ and $E_6^{(2)}$. Using this presentation, we construct exact sequences among these principal subspaces, and use these exact sequences to obtain recursions satisfied by graded dimensions of the principal subspaces. Solving these recursions, we obtain the graded dimensions of the principal subspaces.
We construct combinatorial bases of principal subspaces of standard modules of level $k \geq 1$ with highest weight $kΛ_0$ for the twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$. Using these bases we directly calculate characters of principal subspaces.
We construct quasi-particle bases of principal subspaces of standard modules $L(Λ)$, where $Λ=k_0Λ_0+k_jΛ_j$, and $Λ_j$ denotes the fundamental weight of affine Lie algebras of type $B_l^{(1)}$, $C_l^{(1)}$, $F_4^{(1)}$ or $G_2^{(1)}$ of level one. From the given bases we find characters of principal subspaces.
Given a collection of $t$ subspaces in an $n$-dimensional $\mathbb{K} $-vector space $W$ we can associate to them $t$ vanishing ideals in the symmetric algebra $\mathcal{S}(W^*) = \mathbb{K}[x_1,x_2,\dots,x_n]$. As a subspace is defined by a set of linear equations, its vanishing ideal is generated by linear forms so it is a linear ideal. Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of $t$ linear ideals is equal to $t$. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of $t$ linear ideals is at most $t$ and they also showed…
In the paper we propose a proof of Reeder's Conjecture on the graded multiplicities of small representations in the exterior algebra $Λ$g for the simple Lie algebras of type B and C.
We describe the fermionic and bosonic Fock representation of the Lie super-algebra of endomorphisms of the exterior algebra of the ${\mathbb Q}$-vector space of infinite countable dimension, vanishing at all but finitely many basis elements. We achieve the goal by exploiting the extension of the Schubert derivations to the Fermionic Fock space.
Symplectic spinor valued forms and invariant operators acting between them
在具有亚辛结构的流形上，将取值于（Kostant）辛旋量丛的外微分形式分解为不变子空间。投影到这些不变子空间的共变导数相关的挠自由辛连接。
Lie superalgebra structures in H*(g; g)
在一个给定的流形M上，Nijenhuis括号使向量值微分形式的超空间成为一个李超代数，这个李超代数可以被解释为外微分的中心化子，外微分被认为是与M上的de Rham丛相关联的超流形上的一个向量场。一个类似的括号引入了DG李超代数在上链和上同调的超空间中的结构，系数在任何李超代数的伴随模中。我们使用一个基于Mathematica的包SuperLie（已经证明在各种问题中有用）来明确描述这些李超代数的一些简单的有限维李超代数和它们的"相对“（非平凡的中心扩展或导子代数的考虑简单的）。
On the Spectrum of Exterior Algebra, and Generalized Exponents of Small Representations
本文研究了复单李代数外代数中的不可约表示，证明了B、C、D型李代数中特定不可约表示的出现，提出了Kostant猜想的类似猜想，并给出了基础小表示广义指数的闭式表达式及特殊协变量模结果的推广。
Vertex-algebraic structure of the principal subspaces of level one modules for the untwisted affine Lie algebras of types A,D,E
推广我们以前的一些工作，我们证明了自然的介绍，一级标准模的无扭仿射李代数的类型A，D和E的主子空间，以及某些相关的空间。作为结果，我们得到了这些空间的（多）分次维数的递归（q-差分方程）的一个标准完备集，并导出了它们的分次维数。我们的方法是基于顶点算子代数理论中的交织算子。
Extensions of super Lie algebras
我们研究了一个给定的超李代数的（非交换）扩张，确定了一个上同调障碍的存在性，平行于已知的李代数。类比设置协变的外部衍生物，曲率，和微分几何中的比安奇身份的阐述。
We provide an observation relating several known and conjectured $q$-series identities to the theory of principal subspaces of basic modules for twisted affine Lie algebras. We also state and prove two new families of $q$-series identities. The first family provides quadruple sum representations for Nandi's identities, including a manifestly positive representation for the first identity. The second is a family of new mod 10 identities connected with principal characters of level 4 integrable, highest-weight modules of $\mathrm{D}_4^{(3)}$.
In 1998, Liebeck and Seitz introduced a constant $t(G)$, dependent on the root system of a reductive algebraic group $G$ and proved that if $x$ is a semisimple element of order greater than $t(G)$ in $G$ then there exists an infinite subgroup of $G$ stabilizing the same subspaces of $L(G)$ as $x$. The values for $t(G)$ are $12$, $68$, $124$ and $388$ for $G=G_2,F_4,E_6,E_7$ respectively. In this paper we obtain a similar result for these groups and the minimal module $V_{\mathrm{min}}$, obtaining significantly smaller numbers, namely $4$, $18$, $27$ and $75$ respectively (with some small condi…
Let $(π, \mathcal{H})$ be a strongly continuous unitary representation of a 1-connected Lie group $G$ such that the Lie algebra $\mathfrak{g}$ of $G$ is generated by the positive cone $C_π:= \{x \in \mathfrak{g} : -i\partial π(x) \geq 0\}$ and an element $h$ for which the adjoint representation of $h$ induces a 3-grading of $\mathfrak{g}$. Moreover, suppose that $(π, \mathcal{H})$ extends to an antiunitary representation of the extended Lie group $G_τ:= G \rtimes \{\mathbf{1}, τ_G\}$, where $τ_G$ is an involutive automorphism of $G$ with $\mathbf{L}(τ_G) = e^{iπ\mathrm{ad} h}$. In a recent wor…
For a non-abelian Lie algebra $L$ of dimension $n$ with the derived subalgebra of dimension $m$ , the first author earlier proved that the dimension of its Schur multiplier is bounded by $\frac{1}{2}(n+m-2)(n-m-1)+1$. In the current work, we give some new inequalities on the exterior square and the Schur multiplier of Lie algebras and then we obtain the class of all nilpotent Lie algebras which attains the above bound. Moreover, we also improve this bound as much as possible.
Stabilizers of Subspaces under Similitudes of the Klein Quadric, and Automorphisms of Heisenberg Algebras
本文确定了任意域（包括特征2情形）上2类小维幂零李代数的自同构群及其轨道。
Let G be a Lie group with Lie algebra $\mathfrak{g}$, $h \in \frak{g}$ an element for which the derivation ad(h) defines a 3-grading of $\mathfrak{g}$ and $τ_G$ an involutive automorphism of G inducing on $\mathfrak{g}$ the involution $e^{πi ad(h)}$. We consider antiunitary representations $U$ of the Lie group $G_τ= G \rtimes \{e,τ_G\}$ for which the positive cone $C_U = \{ x \in \mathfrak{g} : -i \partial U(x) \geq 0\}$ and $h$ span $\mathfrak{g}$. To a real subspace E of distribution vectors invariant under $exp(\mathbb{R} h)$ and an open subset $O \subseteq G$, we associate the real subspac…
Let V be a standard subspace in the complex Hilbert space H and U : G \to U(H) be a unitary representation of a finite dimensional Lie group. We assume the existence of an element h in the Lie algebra of G such that U(exp th) is the modular group of V and that the modular involution J_V normalizes U(G). We want to determine the semigroup $S_V = \{ g\in G : U(g)V \subseteq V\}.$ In previous work we have seen that its infinitesimal generators span a Lie algebra on which ad h defines a 3-grading, and here we completely determine the semigroup S_V under the assumption that ad h defines a 3-grading…
Nets of real subspaces on homogeneous spaces and Algebraic Quantum Field Theory
本文通过将AQFT中时空区域参数化的算子代数网公理转化为实子空间网公理，从李群几何与表示论视角建立了李群幺正表示与局部代数网的联系。
Combinatorial bases of Feigin-Stoyanovsky's type subspaces of level 1 standard modules for $\tilde{\mathfrak sl}(\ell+1,\C)$
设$\tilde{\mathfrak g}$是$A_\ell^{(1)}$型仿射李代数.假设我们给定相应的简单有限维李代数${\mathfrak g}={\mathfrak g}_{-1}\oplus{\mathfrak g}_0 \oplus {\mathfrak g}_1$的$\mathbb Z$ -阶化，那么我们也有仿射李代数$$\tilde{\mathfrak g}=\tilde{\mathfrak g}_{-1} \oplus \tilde{\mathfrak g}_0 \oplus \tilde{\mathfrak g}_1.$$的诱导$\mathbb Z$ -阶化设$L(Λ)$是一个标准的1级模。Feigin-Stoyanovsky的型子空间$W(Λ)$是$\tilde{\mathfrak g}_1$ -由最高权向量$v_Λ$，$$W(Λ)=U(\tilde{\mathfrak g}_1)\cdot v_Λ\subset L(Λ).$$生成的$L(Λ)$的子模。利用交织算子的系数，归纳地证明了生成集的线性无关性。获得$L(Λ)$的基作为$W(Λ)$的基的“归纳极限"。
On the derivation algebra of the free Lie algebra and trace maps
本文计算了自由群和自由亚交换群的自同构群的Johnson同态的象的不可约分解。本文确定了Chen李代数的导子代数代数为李代数的Abelianization，并证明了该Abelianization是由度1部分和Morita迹映射给出的.我们还考虑了自由幂零群的自同构群的扭上同调群。我们证明了外积的迹映射定义了它的一个非平凡的扭曲的第二上同调类。
Extensions of Lie algebras
我们回顾了一个给定的李代数的（非交换）扩张，确定了一个三维上同调障碍的存在的扩展。一个惊人的类比设置协变的外部衍生物，曲率，和微分几何中的比安奇身份的阐述。在新版本中添加了参考文献：大多数结果都是已知的。因此，这篇论文不会提交给期刊，它可以被视为综述论文。
Rindler wedges are fundamental localization regions in AQFT. They are determined by the one-parameter group of boost symmetries fixing the wedge. The algebraic canonical construction of the free field provided by Brunetti-Guido-Longo (BGL) arises from the wedge-boost identification, the BW property and the PCT Theorem. In this paper we generalize this picture in the following way. Firstly, given a $\mathbb Z_2$-graded Lie group we define a (twisted-)local poset of abstract wedge regions. We classify (semisimple) Lie algebras supporting abstract wedges and study special wedge configurations. Th…
Versal Deformations of Three Dimensional Lie Algebras as L-infinity Algebras
我们考虑0.00000的|3维L-无穷代数，它精确地对应于普通的（非分次的）3维李代数。这种代数在C上的分类是众所周知的，尽管我们将使用将它们视为L-无穷代数的方法来推导这种分类。由于三维奇向量空间的对称代数只含有外次数小于或等于3的项，因此可以完全地构造奇向量空间的对称变形.我们给出了一个表征的模空间的李代数使用变形理论作为指导，以了解图片。
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