Céline Righi
We extend the results of Cellini-Papi on the characterizations of nilpotent and abelian ideals of a Borel subalgebra to parabolic subalgebras of a simple Lie algebra. These characterizations are given in terms of elements of the affine Weyl group and faces of alcoves. In the case of a parabolic subalgebra of a classical Lie algebra, we give formulas for the number of these ideals.
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
Using GAP4, we determine the number of ad-nilpotent and abelian ideals of a parabolic subalgebra of a simple Lie algebra of exceptional types E, F or G.
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.
We determine the Lie subalgebra $\mathfrak{g}_{nil}$ of a Borcherds symmetrizable generalized Kac-Moody Lie algebra $\mathfrak{g}$ generated by $ad$-locally nilpotent elements and show that it is `essentially' the same as the Levi subalgebra of $\mathfrak{g}$ with its simple roots precisely the real simple roots of $\mathfrak{g}$.
Simple modules for untwisted affine Lie algebras induced from nilpotent loop subalgebras
本文利用幂零循环子代数上一维模的诱导，构造了无扭仿射李代数的大类单模，并证明了其首自扩张空间具有不可数维且通用张量积为单模。
We provide an explicit bijection between the ad-nilpotent ideals of a Borel subalgebra of a simple Lie algebra g and the orbits of \check{Q}/(h+1)\check{Q} under the Weyl group (\check{Q} being the coroot lattice and h the Coxeter number of g). From this result we deduce in a uniform way a counting formula for the ad-nilpotent ideals.
In this paper, we give upper bounds for the index of the quotient of the Borel subalgebra of a simple Lie algebra or its nilpotent radical by an ad-nilpotent ideal. For the nilpotent radical quotient, our bound is a generalization of the formula for the index given by Panov in the type A case. In general, this bound is not exact. Using results from Panov, we show that the upper bound for the Borel quotient is exact in the type $A$ case, and we conjecture that it is exact in general.
This paper is devoted to a detailed study of certain remarkable posets which form a natural partition of all abelian ideals of a Borel subalgebra. Our main result is a nice uniform formula for the dimension of maximal ideals in these posets. We also obtain results on ad-nilpotent ideals which complete the analysis started in \cite{CP2}, \cite{CP3}.
Odd nilpotent element and $\mathfrak{osp}(1|2)$-subalgebra in $\mathfrak{gl}(m|n)$
本文研究$\mathfrak{gl}(m|n)$中奇幂零元嵌入$\mathfrak{osp}(1|2)$-子代数的条件，通过定义超Jordan矩阵并证明充要条件：该元位于仅含奇数阶超Jordan块的超Jordan矩阵轨道中。
We characterize locally semisimple subalgebras $ł$ of $\sl_\infty$, $\so_\infty$, and $\sp_\infty$ which are Levi components of parabolic subalgebras. Given $ł$, we characterize the parabolic subalgebras $\p$ such that $ł$ is a Levi component of $\p$. When the set of such self-normalizing parabolic subalgebras $\p$ is finite, we prove an estimate on its cardinality. We consider various examples which highlight the differences from the case of parabolic subalgebras of finite-dimensional simple Lie algebras.
On adjoint orbits in nilpotent ideals of a Borel subalgebra
本文研究复半单 Lie 代数 Borel 子代数幂零理想中的伴随轨道，证明由确定该最小理想的幂零元组成的集合恰有一个闭 Borel 子群轨道，并用相关极小根刻画其代表元。
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…
Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We collect some general results on the Poisson center of S(g), including some simple criteria regarding its polynomiality, and also on certain Poisson commutative subalgebras of S(g). These facts are then used to finish our earlier work on this subject, i.e. to give an explicit description for the Poisson center of all indecomposable, nilpotent Lie algebras of dimension at most seven. Among other things, we also provide a polynomial, maximal Poisson commutative subalgebra of S(g), enjoying ad…
In the present paper the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra is studied. We set up general conjectures on the construction of the field of invariants and the structure of orbits of maximal dimension. The conjecture is proved for parabolic subalgebras of special types.
In this short note, we investigate the following question of Panyushev : ``Is the sum of the index of a parabolic subalgebra of a semisimple Lie algebra $\mathfrak{g}$ and the index of its nilpotent radical always greater than or equal to the rank of $\mathfrak{g}$?''. Using the formula for the index of parabolic subalgebras conjectured by Tauvel and the author, and proved by Millet-Fauquant and Joseph, we give a positive answer to this question. Moreover, we also obtain a necessary and sufficient condition for this sum to be equal to the rank of $\mathfrak{g}$. This provides new examples of d…
In the paper the algebra of invariants of the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra is studied. We set up a conjecture on the structure of the algebra of invariants. The conjecture is proved for parabolic subalgebras of special types.
In the present paper the field of invariants of the adjoint action of the unitriangular group in the nilradical of any parabolic subalgebra is described.
Let $g$ be a finite-dimensional simple Lie algebra over the complex number field. We classify the homomorphisms between $g$-modules induced from one-dimensional modules of maximal parabolic subalgebras.
We say that a finite dimensional Lie algebra is quasi-reductive if it has a linear form whose stabilizer for the coadjoint representation, modulo the center, is a reductive Lie algebra with a center consisting of semisimple elements. Parabolic subalgebras of a semisimple Lie algebra are not always quasi-reductive (except in types A or C by work of Panyushev). The classification of quasi-reductive parabolic subalgebras in the classical case has been recently achieved in unpublished work of Duflo, Khalgui and Torasso. In this paper, we investigate the quasi-reductivity of biparabolic subalgebras…
Let $\g$ be a locally reductive complex Lie algebra which admits a faithful countable-dimensional finitary representation $V$. Such a Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of $\sl_\infty$, $\so_\infty$, $\sp_\infty$, and finite-dimensional simple Lie algebras. A parabolic subalgebra of $\g$ is any subalgebra which contains a maximal locally solvable (that is, Borel) subalgebra. Building upon work by Dimitrov and the authors of the present paper, we give a general description of parabolic subalgebras of $\g$ in terms of joint stabilizers of taut co…
On the reducibility of scalar generalized Verma modules associated to two-step nilpotent parabolic subalgebras
本文利用Gelfand-Kirillov维数研究非极大型二步幂零抛物子代数对应的标量广义Verma模可约性，限定三类李代数，并将可约点表示在二维复平面中。
This paper gives a classification of parabolic subalgebras of simple Lie algebras over $\CC$ that are complexifications of parabolic subalgebras of real forms for which Lynch's vanishing theorem for generalized Whittaker modules is non-vacuous. The paper also describes normal forms for the admissible characters in the sense of Lynch (at least in the quasi-split cases) and analyzes the important special case when the parabolic is defined by an even embedded TDS (three dimensional simple Lie algebra).
In this paper the field of invariants for the adjoint action of the Borel group in the nilradical of a parabolic subalgebra is studied. We construct the set of B-invariant rational functions generating the field of invariants.
By using a free monoid of operators on the set of compositions (resp. pairs of compositions), we establish in this paper a bijective correspondence between Frobenius standard parabolic (resp. seaweed) subalgebras and certain elements of this monoid. We prove via this correspondence a conjecture of one of the authors on the number of Frobenius standard parabolic (resp. seaweed) subalgebras of $\mathrm{sl}_{n}(\Bbbk )$ associated to compositions (resp. pairs of compositions) with $n-t$ parts (resp. parts in total).
Let p be a maximal truncated parabolic subalgebra of a simple Lie Algebra. It was shown in many cases that the Poisson centre Y(p) is a polynomial algebra. We construct a slice for the coadjoint action of p, thus extending a theorem of Kostant. The role of the principal sl_2-triple is played by an adapted pair.
For any simple Lie algebra $\mathfrak{g}$ and an element $μ\in\mathfrak{g}^*$, the corresponding commutative subalgebra $\mathcal{A}_μ$ of $\mathcal{U}(\mathfrak{g})$ is defined as a homomorphic image of the Feigin-Frenkel centre associated with $\mathfrak{g}$. It is known that when $μ$ is regular this subalgebra solves Vinberg's quantisation problem, as the graded image of $\mathcal{A}_μ$ coincides with the Mishchenko-Fomenko subalgebra $\overline{\mathcal{A}}_μ$ of $\mathcal{S}(\mathfrak{g})$. By a conjecture of Feigin, Frenkel and Toledano Laredo, this property extends to an arbitrary eleme…
A (vector space) basis B of a Lie algebra is said to be very nilpotent if all the iterated brackets of elements of B are nilpotent. In this note, we prove a refinement of Engel's Theorem. We show that a Lie algebra has a very nilpotent basis if and only if it is a nilpotent Lie algebra. When g is a semisimple Lie algebra, this allows us to define an ideal of S((g^n)^*)^G whose associated algebraic set in g^n is the set of n-tuples lying in a same Borel subalgebra.
We study invariants of the adjoint action of the unipotent group in the nilradical of a parabolic subalgebra of types Bn, Cn, Dn. We introduce the notion of expanded base in the set of positive roots and construct an invariant for every root of the expanded base. We prove that these invariants are algebraically independent. We also give an estimate of the transcendence degree of the field of invariants.
Let $G$ be a simply connected semisimple algebraic group with Lie algebra $\mathfrak g$, let $G_0 \subset G$ be the symmetric subgroup defined by an algebraic involution $σ$ and let $\mathfrak g_1 \subset \mathfrak g$ be the isotropy representation of $G_0$. Given an abelian subalgebra $\mathfrak a$ of $\mathfrak g$ contained in $\mathfrak g_1$ and stable under the action of some Borel subgroup $B_0 \subset G_0$, we classify the $B_0$-orbits in $\mathfrak a$ and we characterize the sphericity of $G_0 \mathfrak a$. Our main tool is the combinatorics of $σ$-minuscule elements in the affine Weyl …
We study ad-nilpotent ideals of a parabolic subalgebra of a simple Lie algebra. Any such ideal determines an antichain in a set of positive roots of the simple Lie algebra. We give a necessary and sufficient condition for an antichain to determine an ad-nilpotent ideal of the parabolic. We write down all such antichains for the classical simple Lie algebras and in particular recover the results of D. Peterson. In section 2 of the paper we study the unique ideal in a parabolic which is irreducible as a module for the reductive part and give several equivalent statements that are satisfied by th…
In this paper, we shall recall and arrange the relationship between hyperbolic elements and parabolic subalgebras at first. And then, we shall classify all the compatible parabolic subalgebras containing a $τ$-stable Borel subalgebra for complex symmetric pairs $(\mathfrak{g},\mathfrak{g}^τ)$, where $τ$ is a involutive automorphism of $\mathfrak{g}$.
We consider the conjugation-action of an arbitrary upper-block parabolic subgroup of ${\rm GL}_n(\mathbf{C})$ on the variety of $x$-nilpotent complex matrices and translate it to a representation-theoretic context. We obtain a criterion as to whether the action admits a finite number of orbits and specify a system of representatives for the orbits in the finite case of $2$-nilpotent matrices. Furthermore, we give a set-theoretic description of their closures and specify the minimal degenerations in detail for the action of the Borel subgroup. We show that in all non-finite cases, the correspon…
In this paper the authors introduce a class of parabolic subalgebras for classical simple Lie superalgebras associated to the detecting subalgebras introduced by Boe, Kujawa and Nakano. These parabolic subalgebras are shown to have good cohomological properties governed by the Bott-Borel-Weil theorem involving the zero component of the Lie superalgebra in conjunction with the odd roots. These results are later used to verify an open conjecture given by Boe, Kujawa and Nakano pertaining to the equality of various support varieties.
Parabolic subalgebras $\frak{p}$ of semisimple Lie algebras define a $\Bbb{Z}$-grading of the Lie algebra. If there exists a nilpotent element in the first graded part of $\frak{g}$ on which the adjoint group of $\frak{p}$ acts with a dense orbit, the parabolic subalgebra is said to be nice. The corresponding nilpotent element is also called admissible. Nice parabolic subalgebras of simple Lie algebras have been classified. In the case of Borel subalgebras a Richardson element of $\frak{g}_1$ is exactly one that involves all simple root spaces. It is however difficult to write down such nilpot…
Let $G$ be a connected semisimple Lie group with a finite center, with a maximal compact subgroup $K$. There are two general methods to construct representations of $G$, namely, ordinary parabolic induction and cohomological parabolic induction. We define Eisenstein integrals relative to cohomological inductions which generalize Flensted-Jensen's fundamental functions for discrete series. They are analogous to Harish-Chandra's Eisenstein integrals related to ordinary inductions. We introduce the notion of Li positivity of a $K$ type in a representation of $G$ which is extremely useful in the s…
It was shown by Cellini and Papi that an ad-nilpotent ideal determines certain element of the affine Weyl group, and that there is a bijection between the ad-nilpotent ideals and the integral points of a simplex with rational vertices. We give a description of the generators of ad-nilpotent ideals in terms of these elements, and show that an ideal has $k$ generators if and only it lies on the face of this simplex of codimension $k$. We also consider two combinatorial statistics on the set of ad-nilpotent ideals: the number of simple roots in the ideal and the number of generators. Considering …
We construct nonlinear oblique projections along subalgebras of nilpotent Lie algebras in terms of the Baker-Campbell-Hausdorff multiplication. We prove that these nonlinear projections are real analytic on every Schubert cell of the Grassmann manifold whose points are the subalgebras of the nilpotent Lie algebra under consideration.
Saturation rank for nilradical of parabolic subalgebras in Type A
本文通过构造有限连通箭图，组合刻画了A型抛物子代数幂零根中Richardson元的中心化子，并证明在温和条件下幂零根的饱和秩等于SL_{n+1}(K)的半单秩。
In this note, we describe some desingularizations of some subvarieties of the cartesian powers of a semisimple Lie algebra of finite dimension.
We use affine W-algebras to quantize Mishchenko-Fomenko subalgebras for centralizers of nilpotent elements in simple Lie algebras under certain assumptions that are satisfied for all cases in type A and all minimal nilpotent cases outside type $E_8$.
Motivated by questions in modular representation theory, Carlson, Friedlander, and the first author introduced the varieties E(r, g) of r-dimensional abelian p-nilpotent subalgebras of a p-restricted Lie algebra g. In this paper, we identify the varieties E(r, g) for a reductive restricted Lie algebra g and r the maximal dimension of an abelian p-nilpotent subalgebra of g.
We study the interaction between the block decompositions of reduced universal enveloping algebras in positive characteristic, the PBW filtration, and the nilpotent cone. We provide two natural versions of the PBW filtration on the block subalgebra $A_λ$ of the restricted universal enveloping algebra $\mathcal{U}_χ(\mathfrak{g})$ and show these are dual to each other. We also consider a shifted PBW filtration for which we relate the associated graded algebra to the algebra of functions on the Frobenius neighbourhood of $0$ in the nilpotent cone and the coinvariants algebra corresponding to $λ$…
Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary field of characteristic zero, which does not rely upon the structure theory of semisimple Lie algebras. Indeed we derive such structure theory, from root systems to the Bruhat decomposition, from the properties of parabolic subalgebras. As well as constructing the Tits building of a …
We study the cohomology with trivial coefficients of Lie algebras L_k of the polynomial vector fields on the line with zero $k$-jet, (k>=1), and the cohomology of the similar subalgebras {L}_k of the polynomial loops algebra $\widetilde{sl}_2$. In both cases we construct the special bases (filtering bases) in the external complexes of these algebras. A spectral sequence based on this construction allows to completely find the cohomology of L_k and {L}_k. We also apply the filtering bases to find the spectral resolution of the Laplace operators for algebras L_1 and L_0, and obtain explicit form…
Tensor Products with Verma Module and Restriction to Parabolic Subalgebra
本文利用张量恒等式，基于模对抛物子代数的限制分解，研究了广义Verma模与范畴O^p中模的张量积分解。证明了该限制存在本质上唯一的不可分解分解，并识别出有限维商与倾斜子模两类直和项，最后给出了sl_2和sl_3中相关模的完整Borel分解及显式计算。
In this paper, we study the commuting variety of symmetric pairs associated to parabolic subalgebras with abelian unipotent radical in a simple complex Lie algebra. By using the ``cascade'' construction of Kostant, we construct a Cartan subspace which in turn provides, in certain cases, useful information on the centralizers of non $\mathfrak{p}$-regular semisimple elements. In the case of the rank 2 symmetric pair $(\mathrm{so}_{p+2},\mathrm{so}_{p}\times \mathrm{so}_{2})$, $p\geq 2$, this allows us to apply induction, in view of previous results of the authors, and reduce the problem of the …
Unique Factorization For Tensor Products of Parabolic Verma Modules
本文针对可对称化 Kac-Moody Lie 代数，证明了抛物 Verma 模张量积及其特征标乘积在特定 Cartan 子代数限制下的唯一分解性质。
The two most fundamental conjectures on the structure of the generic Hecke algebra $\mathcal{H}(W)$ associated with a complex reflection group $W$ state that $\mathcal{H}(W)$ is a free module of rank $|W|$ over its ring of definition, and that $\mathcal{H}(W)$ admits a canonical symmetrising trace. The first conjecture has recently become a theorem, while the second conjecture, known to hold for real reflection groups, has only been proved for some exceptional non-real complex reflection groups (all of rank $2$ but one). The two most fundamental conjectures on the structure of the parabolic He…
Let $G$ be a real linear reductive group and $K$ be a maximal compact subgroup. Let $P$ be a minimal parabolic subgroup of $G$ with complexified Lie algebra $\mathfrak{p}$, and $\mathfrak{n}$ be its nilradical. In this paper we show that: for any admissible finitely generated moderate growth smooth Fréchet representation $V$ of $G$, the inclusion $V_{K}\subset V$ induces isomorphisms $H_{i}(\mathfrak{n},V_{K})\cong H_{i}(\mathfrak{n},V)$ ($i\geq 0$), where $V_{K}$ denotes the $(\mathfrak{g},K)$ module of $K$ finite vectors in $V$. This is called Casselman's comparison theorem. As a consequence…
We show that the Poisson centre of truncated maximal parabolic subalgebras of a simple Lie algebra of type B, D and E_6 is a polynomial algebra. In roughly half of the cases the polynomiality of the Poisson centre was already known by a completely different method. For the rest of the cases, our approach is to construct an algebraic slice in the sense of Kostant given by an adapted pair and the computation of an improved upper bound for the Poisson centre.
Reducibility of Scalar Generalized Verma Modules of Minimal Parabolic Type II
本文研究了例外李代数中最小抛物子代数相关标量广义韦尔玛模的可约性，通过计算盖尔范-基里洛夫维数确定其第一个对角可约点。
According to Kirillov's theory, the construction of a unitary irreducible representation of a nilpotent Lie group requires a precise computation of some polarizing subalgebra subordinated to a linear functional in the linear dual of the corresponding Lie algebra. This important step is generally challenging from a computational viewpoint. In this paper, we provide an algorithmic approach to the construction of the well-known Vergne polarizing subalgebras \cite{Corwin}. The algorithms presented in this paper are specifically designed so that they can be implemented in Computer Algebra Systems. …
In the present paper we continue the project of systematic classification and construction of invariant differential operators for non-compact semisimple Lie groups. This time we make the stress on one of the main building blocks, namely the Verma modules and the corresponding parabolic subalgebras. In particular, we start the study of the relation between the parabolic subalgebras of real semisimple Lie algebras and of their complexification. Two cases are given in more detail: the conformal algebra of 4D Minkowski space-time and the minimal parabolics of classical real semisimple Lie algebra…
Reducibility of scalar generalized Verma modules of minimal parabolic type
本文针对经典复单李代数的最小抛物子代数，通过计算最高权模的Gelfand-Kirillov维数，确定了标量广义Verma模的第一个对角可约点。
Let g be a complex simple Lie algebra and e a nilpotent element in g. To a certain nilpotent subalgebra m attached to e, called an admissible subalgebra of g, we associate an endomorphism algebra H. When m is constructed from a good grading for e, we recover the finite W-algebra associated to e and it is well-known that gr(H) is isomorphic to \C[S] as a graded Poisson algebra where S is the Slodowy slice of e and gr(H) is the graded algebra associated to the Kazhdan filtration. In this paper, we consider the case where g =\sl_{pn}(\C) and e consists of p Jordan blocks all of the same size n. H…
It is known that the variety parametrizing pairs of commuting nilpotent matrices is irreducible and that this provides a proof of the irreducibility of the punctual Hilbert scheme in the plane. We extend this link to the nilpotent commuting variety of parabolic subalgebras of $M\_n(\K)$ and to the punctual nested Hilbert scheme. By this method, we obtain a lower bound on the dimension of these moduli spaces. We characterize the numerical conditions under which they are irreducible. In some reducible cases, we describe the irreducible components and their dimension.
The flag variety of a complex reductive linear algebraic group G is by definition the quotient G/B by a Borel subgroup. It can be regarded as the set of Borel subalgebras of Lie(G). Given a nilpotent element e in Lie(G), one calls Springer fiber the subvariety formed by the Borel subalgebras which contain e. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by C. De Concini, G. Lusztig, and C. Procesi asserts that, when G is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In th…
We consider the conjugation-action of the Borel subgroup of the symplectic or the orthogonal group on the variety of nilpotent complex elements of nilpotency degree $2$ in its Lie algebra. We translate the setup to a representation-theoretic context in the language of a symmetric quiver algebra. This makes it possible to provide a parametrization of the orbits via a combinatorial tool that we call symplectic/orthogonal oriented link patterns. We deduce information about numerology. We then generalize these classifications to standard parabolic subgroups for all classical groups. Finally, our r…
In this paper, using Bourbaki's convention, we consider a simple Lie algebra $\mathfrak g\subset\mathfrak g\mathfrak l_m$ of type B, C or D and a parabolic subalgebra $\mathfrak p$ of $\mathfrak g$ associated with a Levi factor composed essentially, on each side of the second diagonal, by successive blocks of size two, except possibly for the first and the last ones. Extending the notion of a Weierstrass section introduced by Popov to the coadjoint action of the truncated parabolic subalgebra associated with $\mathfrak p$, we construct explicitly Weierstrass sections, which give the polynomial…
In this paper we study gradings on simple Lie algebras arising from nilpotent elements. Specifically, we investigate abelian subalgebras which are degree 1 homogeneous with respect to these gradings. We show that for each odd nilpotent element there always exists canonically associated "strictly" odd nilpotent element, which allows us to reduce our investigations to the latter.
With this paper we start a programme aiming at connecting two vast scientific areas: Jordan algebras and representation theory. Within representation theory, we focus on non-compact, real forms of semisimple Lie algebras and groups as well as on the modern theory of their induced representations, in which a central role is played by the parabolic subalgebras and subgroups. The aim of the present paper and its sequels is to present a Jordan algebraic interpretations of maximal parabolic subalgebras. In this first paper, we confine ourselves to maximal parabolic subalgebras of the non-compact re…
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。