Mouchira Zaiter
Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x, y) in g\timesg such that x and y are nilpotents and are in the same Borel subalgebra. The main result of this paper is that N is a closed and irreducible subvariety of g \times g whose normalization has rational singularities and such that the normalization morphism is bijective.
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Let g be a semisimple Lie algebra of finite dimension. The nullcone N of g is the set of (x,y) in g x g such that x and y are nilpotents and are in the same Borel sualgebra. The main result of this paper is that N is a closed and irreducible subvariety of g x g, its normalisation has rational singularities and its normalization morphism is bijective.
Let $ζ$ be a complex $\ell$th root of unity for an odd integer $\ell>1$. For any complex simple Lie algebra $\mathfrak g$, let $u_ζ=u_ζ({\mathfrak g})$ be the associated "small" quantum enveloping algebra. In general, little is known about the representation theory of quantum groups (resp., algebraic groups) when $l$ (resp., $p$) is smaller than the Coxeter number $h$ of the underlying root system. For example, Lusztig's conjecture concerning the characters of the rational irreducible $G$-modules stipulates that $p \geq h$. The main result in this paper provides a surprisingly uniform answer f…
We start the study of glider representations in the setting of semisimple Lie algebras. A glider representation is defined for some positively filtered ring $FR$ and here we consider the right bounded algebra filtration $FU(\mathfrak{g})$ on the universal enveloping algebra $U(\mathfrak{g})$ of some semisimple Lie algebra $\mathfrak{g}$ given by a fixed chain of semisimple sub Lie algebras $\mathfrak{g}_1 \subset \mathfrak{g}_2 \subset \ldots \subset \mathfrak{g}_n = \mathfrak{g}$. Inspired by the classical representation theory, we introduce so-called Verma glider representations. Their exist…
Orbits and invariants for coisotropy representations
本文研究约化群齐性空间余迷向表示的轨道与不变量，补充了复杂度与秩的已有结果，并建立零锥联系，探讨同调维数及泊松交换子代数。
We study deformations of orbit closures for the action of a connected semisimple group $G$ on its Lie algebra $\mathfrak{g}$, especially when $G$ is the special linear group. The tools we use are on the one hand the invariant Hilbert scheme and on the other hand the sheets of $\mathfrak{g}$. We show that when $G$ is the special linear group, the connected components of the invariant Hilbert schemes we get are the geometric quotients of the sheets of $\mathfrak{g}$. These quotients were constructed by Katsylo for a general semisimple Lie algebra $\mathfrak{g}$; in our case, they happen to be af…
In his paper "A Construction for Coisotropic Subalgebras of Lie Bialgebras", Marco Zambon gave a way to use a long root of a complex semisimple Lie biaglebra $\mathfrak{g}$ to construct a coisotropic subalgebra of $\mathfrak{g}$. In this paper, we generalize Zambon's construction. Our construction is based on the theory of Lagrangian subalgebras of the double $\mathfrak{g}\oplus\mathfrak{g}$ of $\mathfrak{g}$, and our coisotropic subalgebras correspond to torus fixed points in the variety $\mathcal{L}(\mathfrak{g}\oplus\mathfrak{g})$ of Lagrangian subalgebras of $\mathfrak{g}\oplus\mathfrak{g}…
If ${\mathfrak g}$ is a real reductive Lie algebra and ${\mathfrak h} < {\mathfrak g}$ is a subalgebra, then $({\mathfrak g}, {\mathfrak h})$ is called real spherical provided that ${\mathfrak g} = {\mathfrak h} + {\mathfrak p}$ for some choice of a minimal parabolic subalgebra ${\mathfrak p} \subset {\mathfrak g}$. In this paper we classify all real spherical pairs $({\mathfrak g}, {\mathfrak h})$ where ${\mathfrak g}$ is semi-simple but not simple and ${\mathfrak h}$ is a reductive real algebraic subalgebra. The paper is based on the classification of the case where ${\mathfrak g}$ is simple…
Let $σ$ be an involution of a complex semisimple Lie algebra $\mathfrak g$ and $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$ the related $\mathbb Z_2$-grading. We study relations between nilpotent $G_0$-orbits in $\mathfrak g_0$ and the respective $G$-orbits in $\mathfrak g$. If $e\in\mathfrak g_0$ is nilpotent and $\{e,h,f\}\subset\mathfrak g_0$ is an $\mathfrak{sl}_2$-triple, then the semisimple element $h$ yields a $\mathbb Z$-grading of $\mathfrak g$. Our main tool is the combined $\mathbb Z\times\mathbb Z_2$-grading of $\mathfrak g$, which is called a mixed grading. We prove, in particul…
A Lie algebra $\mathfrak{g}_\mathbb{Q}$ over $\mathbb{Q}$ is said to be $\mathbb{R}$-universal if every homomorphism from $\mathfrak{g}_\mathbb{Q}$ to $\mathfrak{gl}(n,\mathbb{R})$ is conjugate to a homomorphism into $\mathfrak{gl}(n,\mathbb{Q})$ (for every $n$). By using Galois cohomology, we provide a short proof of the known fact that every real semisimple Lie algebra has an $\mathbb{R}$-universal $\mathbb{Q}$-form. We also provide a classification of the $\mathbb{R}$-universal Lie algebras that are semisimple.
Cyclic wide subalgebras of semisimple Lie algebras
本文研究半单李代数的循环宽子代数，给出正则Levi可分解子代数为循环宽的判定条件，并在根基为ad幂零子代数的情形下证明该条件是充要的。
For any involution $σ$ of a semisimple Lie algebra $\mathfrak g$, one constructs a non-reductive Lie algebra $\mathfrak k$, which is called a $\mathbb Z_2$-contraction of $\mathfrak g$. In this paper, we attack the problem of describing maximal commutative subalgebras of the Poisson algebra $S(\mathfrak k)$. This is closely related to the study of the coadjoint representation of $\mathfrak k$ and the set, $\mathfrak k^*_{reg}$, of the regular elements of $\mathfrak k^*$. By our previous results, in the context of $\mathbb Z_2$-contractions, the argument shift method provides maximal commutativ…
Let $\mathfrak g$ be a semisimple Lie algebra, $\mathfrak h\subset\mathfrak g$ a reductive subalgebra such that $\mathfrak h^\perp$ is a complementary $\mathfrak h$-submodule of $\mathfrak g$. In 1983, Bogoyavlenski claimed that one obtains a Poisson commutative subalgebra of the symmetric algebra ${\mathcal S}(\mathfrak g)$ by taking the subalgebra ${\mathcal Z}$ generated by the bi-homogeneous components of all $H\in{\mathcal S}(\mathfrak g)^{\mathfrak g}$. But this is false, and we present a counterexample. We also provide a criterion for the Poisson commutativity of such subalgebras ${\mat…
We describe the generators and prove a number of relations for the construction of a planar algebra from the restricted quantum group $\bar{U}_{q}(\mathfrak{sl}_{2})$. This is a diagrammatic description of $End_{\bar{U}_{q}(\mathfrak{sl}_{2})}(X^{\otimes n})$, where $X:=\mathcal{X}^{+}_{2}$ is a two dimensional $\bar{U}_{q}(\mathfrak{sl}_{2})$ module.
We study {\em disemisimple} Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra $\mathfrak{g}$ is disemisimple if and only if its solvable radical coincides with its nilradical and is a prehomogeneous $\mathfrak{s}$-module for a Levi subalgebra $\mathfrak{s}$ of $\mathfrak{g}$. We use the classification of prehomogeneous $\mathfrak{s}$-modules for simple Lie algebras $\mathfrak{s}$ given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend th…
Let $\mathfrak{g}$ be a semisimple complex Lie algebra, and let $W$ be a finite subgroup of $\mathbb{C}$-algebra automorphisms of the enveloping algebra $U(\mathfrak{g})$. We show that the derived category of $U(\mathfrak{g})^W$-modules determines isomorphism classes of both $\mathfrak{g}$ and $W.$ Our proofs are based on the geometry of the Zassenhaus variety of the reduction modulo $p\gg 0$ of $\mathfrak{g}.$ Specifically, we use non-existence of certain étale coverings of its smooth locus
An arbitrary proper parabolic subalgebra ${\mathfrak p}$ of a simple complex Lie algebra ${\mathfrak g}$ induces an embedding ${\mathfrak g}\hookrightarrow \mathbb W_n$, and more generally an embedding ${\mathfrak g}\hookrightarrow \mathbb W_n\otimes \operatorname{End}{} V$, where $\mathbb W_n$ is the Weyl algebra in $n$ variables, $n$ is the dimension of the nilradical of ${\mathfrak p}$, and $V$ is an arbitrary ${\mathfrak p}$-module. We give an elementary proof of this known fact, report on a computer program computing the embeddings, and tabulate exceptional Lie algebra embeddings $G_2 \ho…
Consider a simple Lie algebra $\mathfrak{g}$ and $\overline{\mathfrak{g}}% \subset \mathfrak{g}$ a Levi subalgebra. Two irreducible $\overline{% \mathfrak{g}}$-modules yield isomorphic inductions to $\mathfrak{g}$ when their highest weights coincide up to conjugation by an element of the Weyl group $W$ of $\mathfrak{g}$ which is also a Dynkin diagram automorphism of $% \overline{\mathfrak{g}}$. In this paper we study the converse problem: given two irreducible $\overline{\mathfrak{g}}$-modules of highest weight $μ$ and $ν$ whose inductions to $\mathfrak{g}$ are isomorphic, can we conclude that…
Let $\mathfrak{g}$ be a semisimple complex Lie algebra. Recently, Lusztig simplified the traditional construction of the corresponding Chevalley groups (of adjoint type) using the "canonical basis" of the adjoint representation of~$\mathfrak{g}$. Here, we present a variation of this idea which leads to a new, and quite elementary construction of~$\mathfrak{g}$ itself from its root system. An additional feature of this set-up is that it also gives rise to explicit Chevalley bases of $\mathfrak{g}$.
From the viewpoint of higher homological algebra, we introduce pure semisimple $n$-abelian category, which is analogs of pure semisimple abelian category. Let $Λ$ be an Artin algebra and $\mathcal{M}$ be an $n$-cluster tilting subcategory of $Mod$-$Λ$. We show that $\mathcal{M}$ is pure semisimple if and only if each module in $\mathcal{M}$ is a direct sum of finitely generated modules. Let $\mathfrak{m}$ be an $n$-cluster tilting subcategory of $mod$-$Λ$. We show that $Add(\mathfrak{m})$ is an $n$-cluster tilting subcategory of $Mod$-$Λ$ if and only if $\mathfrak{m}$ has an additive generator…
We describe the second cohomology of a regular semisimple Hessenberg variety by generators and relations explicitly in terms of GKM theory. The cohomology of a regular semisimple Hessenberg variety becomes a module of a symmetric group $\mathfrak{S}_n$ by the dot action introduced by Tymoczko. As an application of our explicit description, we give a formula describing the isomorphism class of the second cohomology as an $\mathfrak{S}_n$-module. Our formula is not exactly the same as the known formula by Chow or Cho-Hong-Lee but they are equivalent. We also discuss its higher degree generalizat…
Wonderful Compactification of a Cartan Subalgebra of a Semisimple Lie Algebra
本文定义了半单李代数Cartan子代数的紧化，证明其为法诺舒伯特簇，并研究了其边界除子、仿射铺砌、Weyl群作用及上同调环。
We prove that the Duflo-Serganova functor $DS_x$ attached to an odd nilpotent element $x$ of $\mathfrak{osp}(m|2n)$ is semisimple, i.e. sends a semisimple representation $M$ of $\mathfrak{osp}(m|2n)$ to a semisimple representation of $\mathfrak{osp}(m-2k|2n-2k)$ where $k$ is the rank of $x$. We prove a closed formula for $DS_x(L(λ))$ in terms of the arc diagram attached to $λ$.
We find the normal form of nilpotent elements in semisimple Lie algebras that generalizes the Jordan normal form in $\mathfrak{sl}_N$, using the theory of cyclic elements.
We prove that if $\mathfrak{s}$ is a solvable Lie algebra of matrices over a field of characteristic 0, and $A\in\mathfrak{s}$, then the semisimple and nilpotent summands of the Jordan-Chevalley decomposition of $A$ belong to $\mathfrak{s}$ if and only if there exist $S,N\in\mathfrak{s}$, $S$ is semisimple, $N$ is nilpotent (not necessarily $[S,N]=0$) such that $A=S+N$.
We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form $h=(h(1),n\dots,n)$ in Lie type $A_{n-1}$. The main result of this paper gives an explicit presentation of the cohomology rings in terms of generators and their relations. Our presentation naturally specializes to Borel's presentation of the cohomology ring of the flag variety and it is compatible with the representation of the symmetric group $\mathfrak{S}_n$ on the cohomology constructed by J. Tymoczko. As a corollary, we also give an explicit presentation of the $\mathfr…
In a previous paper, the authors studied the radical filtration of a Weyl module $Δ_ζ(λ)$ for quantum enveloping algebras $U_ζ(\overset\circ{\mathfrak g})$ associated to a finite dimensional complex semisimple Lie algebra $\overset\circ{\mathfrak g}$. There $ζ^2=\sqrt[e]{1}$ and $λ$ was, initially, required to be $e$-regular. Some additional restrictions on $e$ were required---e.g., $e>h$, the Coxeter number, and $e$ odd. Translation to a facet gave an explicit semisimple series for all quantum Weyl modules with singular, as well as regular, weights. That is, the sections of the filtration are…
In this paper, we determine the blocks of $\mathcal{O}^\mathfrak{p}$ associated with semisimple Lie algebras of type $E$.
Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\mathfrak{b}$ be a Borel subalgebra of $\mathfrak{g}$, $\mathfrak{n}$ be the nilradical of $\mathfrak{b}$, and $U(\mathfrak{n})$ be the universal enveloping algebra of $\mathfrak{n}$. We study primitive ideals of $U(\mathfrak{n})$. Almost all primitive ideals are centrally generated, i.e., are generated by their intersections with the center $Z(\mathfrak{n})$ of $U(\mathfrak{n})$. We present an explicit characterization of the centrally generated primitive ideals of $U(\mathfrak{n})$ in terms of the Dixmier map and the Kostant cascade in…
Let $G$ be a Lie group with real semisimple Lie algebra $\mathfrak{g}$. Further let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be a Cartan decomposition. The maximal compact subgroup $K \subseteq G$ acts on $\mathfrak{p}$ via the adjoint representation and the convex hulls of the resulting orbits are the polar orbitopes. We prove that every polar orbitope is a spectrahedron by giving an explicit representation. In addition we give a new proof for the fact that the faces of a polar orbitope are, up to conjugation, given by the faces of the momentum polytope.
We give a complete classification of Airy structures for finite-dimensional simple Lie algebras over $\mathbb C$, and to some extent also over $\mathbb R$, up to isomorphisms and gauge transformations. The result is that the only algebras of this type which admit any Airy structures are $\mathfrak{sl}_2$, $\mathfrak{sp}_4$ and $\mathfrak{sp}_{10}$. Among these, each admits exactly two non-equivalent Airy structures. Our methods apply directly also to semisimple Lie algebras. In this case it turns out that the number of non-equivalent Airy structures is countably infinite. We have derived a num…
We classify the solvable subalgebras, semisimple subalgebras, and Levi decomposable subalgebras of $\mathfrak{so}(4,\mathbb{C})$, up to inner automorphism. By Levi's Theorem, this is a full classification of the subalgebras of $\mathfrak{so}(4,\mathbb{C})$.
Haar measure for totally positive split real semisimple quantum groups of classical type
本文针对经典型分裂实半单李群，构造了积分其量子包络代数Borel部分的局部紧量子群，计算了其Haar权与模数据，并将其Drinfeld双视为全正半群函数代数的解析量子化。
Non-semisimple topological field theory and $\widehat{Z}$-invariants from $\mathfrak{osp}(1 \vert 2)$
本文基于 Lie 超代数 osp(1|2) 构造非半单拓扑场论，并建立了与物理中 Z-不变量的关联。
Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{t}$ its Cartan subalgebra and $W$ the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel-Moore homology of certain affine Springer fibers for $\mathfrak{g}$ and the global sections of a bundle related to a Procesi bundle on the smooth locus of a partial resolution of $(\mathfrak{t}\oplus \mathfrak{t}^*)/W$. We deduce some applications of our isomorphism including a conditional application to the center of the small quantum group. Our main method is to compare certain bimo…
Let $\mathfrak{g}$ be a semisimple Lie algebra over $\mathbb{C}$ having rank $l$ and let $V=L(λ)$ be an irreducible finite-dimensional $\mathfrak{g}$-module having highest weight $λ.$ Computations of weight multiplicities in $V,$ usually based on Freudenthal's formula, are in general difficult to carry out in large ranks or for $λ$ with large coefficients (in terms of the fundamental weights). In this paper, we first show that in some situations, these coefficients can be "lowered" in order to simplify the calculations. We then investigate how this can be used to improve the aforementioned for…
For a semisimple Lie algebra defined over a discrete valuation ring with field of fractions $K$, we prove that any primitive ideal with rational central character in the affinoid enveloping algebra, $\widehat{U(\mathfrak{g})_{K}},$ is the annihilator of an affinoid highest weight module.
For a semisimple algebraic group $G$ of adjoint type with Lie algebra $\mathfrak g$ over the complex numbers, we establish a bijection between the set of closed orbits of the group $G \ltimes \mathfrak g^{\ast}$ acting on the variety of Lagrangian subalgebras of $\mathfrak g \ltimes \mathfrak g^{\ast}$ and the set of abelian ideals of a fixed Borel subalgebra of $\mathfrak g$. In particular, the number of such orbits equals $2^{\text{rk} \mathfrak g}$ by Peterson's theorem on abelian ideals.
Let $B$ be a Borel subgroup of a semisimple algebraic group $G$, and let $\mathfrak a$ be an abelian ideal of $\mathfrak b=Lie(B)$. The ideal $\mathfrak a$ is determined by certain subset $Δ_{\mathfrak a}$ of positive roots, and using $Δ_{\mathfrak a}$ we give an explicit classification of the $B$-orbits in $\mathfrak a$ and $\mathfrak a^*$. Our description visibly demonstrates that there are finitely many $B$-orbits in both cases. We also describe the Pyasetskii correspondence between the $B$-orbits in $\mathfrak a$ and $\mathfrak a^*$ and the invariant algebras $\Bbbk[\mathfrak a]^U$ and $\B…
The Iwasawa decomposition $\mathfrak{g}=\mathfrak{k}_0 \oplus \hat{\mathfrak{a}}_0 \oplus \mathfrak{n}_0$ of the real semisimple Lie algebra $\mathfrak{g}_0$ comes from its Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{p}_0$. Then we get $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{b}_0$ where $\mathfrak{b}_0=\hat{\mathfrak{a}}_0 \oplus \mathfrak{n}_0$. The question of knowing if the index were additive in the decomposition $\mathfrak{g}_0=\mathfrak{k}_0 \oplus \mathfrak{b}_0$ goes back M. Raïs \cite{Rais}. In \cite{Moreau3}, I wrote that the index always is additive …
Pro-$p$ Iwahori-Hecke modules in semisimple rank one and singularity categories
本文研究了p-进Iwahori-Hecke代数的同调范畴，并将其与特定方案的奇点范畴联系起来，揭示了GL₂情况下的等价关系及SL₂和PGL₂情况下的显式描述。
Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $θ$ an involutive automorphism of $\mathfrak{g}$. According to G. Letzter, S. Kolb and M. Balagović the fixed-point subalgebra $\mathfrak{k} = \mathfrak{g}^θ$ has a quantum counterpart $B$, a coideal subalgebra of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ possessing a universal K-matrix $\mathcal{K}$. The objects $θ$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of Satake diagrams. In the present work we extend this construction to generalized Satake diagrams, combinatorial data firs…
Let $\mathfrak a$ be an algebraic Lie subalgebra of a simple Lie algebra $\mathfrak g$ with index $\mathfrak a \leq \rank \mathfrak g$. Let $Y(\mathfrak a)$ denote the algebra of $\mathfrak a$ invariant polynomial functions on $\mathfrak a^*$. An algebraic slice for $\mathfrak a$ is an affine subspace $η+V$ with $η\in \mathfrak a^*$ and $V \subset \mathfrak a^*$ a subspace of dimension index $\mathfrak a$ such that restriction of function induces an isomorphism of $Y(\mathfrak a)$ onto the algebra $R[η+V]$ of regular functions on $η+V$. Slices have been obtained in a number of cases through th…
A nilpotent orbit $O$ of a complex semisimple Lie algebra $\mathfrak{g}$ has finite fundamental group. Associated with an etale cover of $O$, we have a finite cover of the closure $\bar{O}$ of $O$. In this article we consider the finite cover X associated with the universal cover of a nilpotent orbit $O$ of a classical simple Lie algebra $\mathfrak{g}$. We construct explicitly a Q-factorial terminalization of X in a group theoretic way.
Representations having vectors fixed by a Levi subgroup
本文分类半单实李代数表示中含极大分裂环面中心化子固定非零向量的情形，并改进 $\mathfrak{g}$-标准 Young 表模型。
In analogy to the theory of nilpotent orbit in finite-dimensional semisimple Lie algebras, it is known that the principal $\mathfrak{sl}_2$ subalgebras can be constructed in hyperbolic Kac-Moody Lie algebras. We obtained a series of $\mathfrak{sl}_2$ subalgebras in rank 2 symmetric hyperbolic Kac-Moody Lie algebras by extending the aforementioned construction. We present this result and also discuss $\mathfrak{sl}_2$ modules obtained by the action of the $\mathfrak{sl}_2$ subalgebras on the original Lie algebras.
Let $\mathfrak a$ be an algebraic Lie algebra. An adapted pair for $\mathfrak a$ is pair $(h,η)$ consisting of an ad-semisimple element of $h \in \mathfrak a$ and a regular element of $η\in \mathfrak a^*$ satisfying $(ad \ h)η=-η$. An adapted pair $(h,η)$ is said to satisfy integrality if $ad \ h$ has integer eigenvalues on $\mathfrak a$. Integrality is shown to hold for any Frobenius Lie algebra which is a biparabolic subalgebra of a semisimple Lie algebra; but may fail in general. Call $\mathfrak a$ regular if there are no proper semi-invariant polynomial functions on $\mathfrak a^*$ and if …
Building on Seidel-Solomon's fundamental work, we define the notion of a $\mathfrak{g}$-equivariant Lagrangian brane in an exact symplectic manifold $M$ where $\mathfrak{g} \subset SH^1(M)$ is a sub-Lie algebra of the symplectic cohomology of $M$. When $M$ is a (symplectic) mirror to an (algebraic) homogeneous space $G/P$, homological mirror symmetry predicts that there is an embedding of $\mathfrak{g}$ in $SH^1(M)$. This allows us to study a mirror theory to classical constructions of Borel-Weil and Bott. We give explicit computations recovering all finite dimensional irreducible representati…
Narrow, wide, and $λ$-wide regular subalgebras of semisimple Lie algebras
本文研究半单李代数正则子代数的宽与窄性质，给出了$λ$-宽和宽的充分必要条件，并证明对$\mathfrak{sl}_{n+1}$的正则子代数只需考察伴随表示即可判定宽性，且其正则子代数必为窄或宽。
Let $\mathfrak{g}$ be a complex semisimple Lie algebra. For a regular element $x$ in $\mathfrak{g}$ and a Hessenberg space $H\subseteq \mathfrak{g}$, we consider a regular Hessenberg variety $X(x,H)$ in the flag variety associated with $\mathfrak{g}$. We take a Hessenberg space so that $X(x,H)$ is irreducible, and show that the higher cohomology groups of the structure sheaf of $X(x,H)$ vanish. We also study the flat family of regular Hessenberg varieties, and prove that the scheme-theoretic fibers over the closed points are reduced. We include applications of these results as well.
Let $\mathbb{F}$ be a field of characteristic zero and let $\mathfrak{g}$ be a non-zero finite-dimensional split semisimple Lie algebra with root system $Δ$. Let $Γ$ be a finite set of integral weights of $\mathfrak{g}$ containing $Δ$ and $\{0\}$. Following [2,10], we say that a Lie algebra $L$ over $\mathbb{F}$ is \emph{generalized root graded}, or more exactly $(Γ,\mathfrak{g})$-\emph{graded}, if $L$ contains a semisimple subalgebra isomorphic to $\mathfrak{g}$, the $\mathfrak{g}$-module $L$ is the direct sum of its weight subspaces $L_α$ ($α\inΓ$) and $L$ is generated by all $L_α$ with $α\n…
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…
Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus\mathfrak{g}_{\bar{1}}$ be a finite-dimensional simple Lie superalgebra of type $D(2,1;α)$, $G(3)$ or $F(4)$ over $\mathbb{C}$. Let $G$ be the simply connected semisimple algebraic group over $\mathbb{C}$ such that $\mathrm{Lie}(G)=\mathfrak{g}_{\bar{0}}$. Suppose $e\in\mathfrak{g}_{\bar{0}}$ is nilpotent. We describe the centralizer $\mathfrak{g}^{e}$ of $e$ in $\mathfrak{g}$ and its centre $\mathfrak{z}(\mathfrak{g}^{e})$ especially. We also determine the labelled Dynkin diagram for $e$. We prove theorems relating the dimension of $\left(\mathfra…
Essential Semigroups and Branching Rules
本文针对有限维半单复李代数及其半单子代数的分支规则求解问题，利用本质半群及其子半群描述分支规则，提出有限生成子半群情形下的求解算法，并给出多组李代数对的分支规则示例。
We remove the semisimple condition in the mod-$p$ local-global compatibility result of arXiv:2106.10674. Namely, assuming flatness of $π_{\mathfrak{m}}^\vee$ and $\overlineσ_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ being multiplicity free, we prove that $H^{n-1}_{\text{ét}}(\mathbb{P}_{\mathbb{C}_p}^{n-1}, \mathcal{F}_{π_{\mathfrak{m}}[\mathfrak{m}]})$ determines $\overlineσ_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ uniquely. We give remarks on our assumptions at the end of this note.
This article classifies the real forms of Lie Superalgebra by Vogan diagrams, developing Borel and de Seibenthal theorem of semisimple Lie algebras for Lie superalgebras. A Vogan diagram is a Dynkin diagram of triplet $(\mathfrak{g}_{C},\mathfrak{h_{\bar{0}}},\triangle^{+})$, where $\mathfrak{g}_{C}$ is a real Lie superalgebra, $\mathfrak{h_{\bar{0}}}$ cartan subalgebra, $\triangle^{+}$ positive root system. Although the classification of real forms of contragradient Lie superalgebras is already done. But our method is a quicker one to classify.
The non-semisimple Kazhdan-Lusztig category for affine $\mathfrak{sl}_2$ at admissible levels
本文证明容许水平下仿射$\frak{sl}_2$的Kazhdan-Lusztig范畴是辫张量范畴，建立其泛性质，证明弱对应并构造量子Drinfeld-Sokolov约化的张量范畴版本。
Given a complex semisimple Lie algebra ${\mathfrak g}$ and a commutative ${\mathbb C}$-algebra $A$, let ${\mathfrak g}[A] = {\mathfrak g} \otimes A$ be the corresponding generalized current algebra. In this paper we explore questions involving the computation and finite-dimensionality of extension groups for finite-dimensional ${\mathfrak g}[A]$-modules. Formulas for computing $\operatorname{Ext}^{1}$ and $\operatorname{Ext}^{2}$ between simple ${\mathfrak g}[A]$-modules are presented. As an application of these methods and of the use of the first cyclic homology, we completely describe $\oper…
In this short note we show that the Bruhat cells in real normal forms of semisimple Lie algebras enjoy the same property as their complex analogs: for any two elements $w$, $w'$ in the Weyl group $W(\mathfrak g)$, the corresponding real Bruhat cell $X_w$ intersects with the dual Bruhat cell $Y_{w'}$ iff $w\prec w'$ in the Bruhat order on $W(\mathfrak g)$. Here $\mathfrak g$ is a normal real form of a semisimple complex Lie algebra $\mathfrak g_\mathbb C$. Our reasoning is based on the properties of the Toda flows rather than on the analysis of the Weyl group action and geometric considerations…
Let $B$ be a Borel subgroup of a semisimple algebraic group $G$ and let $\mathfrak m$ be an abelian nilradical in $\mathfrak b={\rm Lie} (B)$. Using subsets of strongly orthogonal roots in the subset of positive roots corresponding to $\mathfrak m$, D. Panyushev \cite{Pan} gives in particular classification of $B-$orbits in $\mathfrak m$ and ${\mathfrak m}^*$ and states general conjectures on the closure and dimensions of the $B-$orbits in both $\mathfrak m$ and ${\mathfrak m}^*$ in terms of involutions of the Weyl group. Using Pyasetskii correspondence between $B-$orbits in $\mathfrak m$ and …
Tensor triangular geometry as introduced by Balmer is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category. In this paper we provide a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the Balmer spectrum. For a classical Lie superalgebra ${\mathfrak g}={\mathfrak g}_{\bar{0}}\oplus {\mathfrak g}_{\bar{1}}$, we construct a Zariski space from a detecting subalgebra of ${\mathfrak g}$ and demonstrate that this topological space governs the tensor triangular geometry f…
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