Christof Geiss, Bernard Leclerc, Jan Schröer
We give a geometric construction of the Verma modules of a symmetric Kac-Moody Lie algebra in terms of constructible functions on the varieties of nilpotent finite-dimensional modules of the corresponding preprojective algebra.
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Associated graded modules of mixed Hodge modules on quivers and preprojective K-theoretic Hall algebra
通过取相伴分次模研究箭图模空间上混合霍奇模与预投射代数的 K - 理论霍尔代数间的联系，还证明了取相伴分次模的函子与反射函子可交换。
We study minimal degenerations between preprojective modules over wild quivers. Asymptotic properties of such degenerations are studied, with respect to codimension and numbers of indecomposable direct summands. We provide families of minimal disjoint degenerations of arbitrary codimension for almost all wild quivers and show that no such examples exist in the remaining cases.
For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Geiß~ et al. (2016) introduced a generalized preprojective algebra $Π(C, D)$. We study tilting modules and support $τ$-tilting modules for the generalized preprojective algebra $Π(C, D)$ and show that there is a bijection between the set of all cofinite tilting ideals of $Π(C,D)$ and the corresponding Weyl group $W(C)$ provided that $C$ has no component of Dynkin type. When $C$ is of Dynkin type, we also establish a bijection between the set of all basic support $τ$-tilting $Π(C,D)$-modules and the corresponding Weyl group $…
The work of the first author on the moment map for representations of quivers included a classification of the possible dimension vectors of simple modules for deformed preprojective algebras. That classification was later used to solve an additive analogue of the Deligne-Simpson problem. The last step in the proof of the classification involved some general position arguments; here we give a new approach which avoids such arguments.
For a symmetrizable GCM $C$ and its symmetrizer $D$, Geiss-Leclerc-Schröer [Invent. Math. 209 (2017)] has introduced a generalized preprojective algebra $Π$ associated to $C$ and $D$, that contains a class of modules, called locally free modules. We show that any basic support $τ$-tilting $Π$-module is locally free and gives a classification theorem of torsion-free classes in $\operatorname{\mathbf{rep}}Π$ as the generalization of the work of Mizuno [Math. Z. 277 (2014)].
The irreducible components of the variety of all modules over the preprojective algebra and MV cycles both index bases of the universal enveloping algebra of the positive part of a semisimple Lie algebra canonically. To relate these two objects Baumann and Kamnitzer associate a cycle in the affine Grassmannian for a given module. It is conjectured that the ring of functions of the T-fixed point subscheme of the associated cycle is isomorphic to the cohomology ring of the quiver Grassmannian of the module. I give a proof of part of this conjecture. Given this conjecture, I give a proof of the r…
In this paper we face the study of the representations of the exceptional Lie superalgebra E(5,10). We recall the construction of generalized Verma modules and give a combinatorial description of the restriction to sl_5 of the Verma module induced by the trivial representation. We use this description to classify morphisms between Verma modules of degree one, two and three proving in these cases a conjecture given by Rudakov. A key tool is the notion of dual morphism between Verma modules.
Computing the extensions between Verma modules is in general a very difficult problem. Using Soergel bimodules, one can construct a graded version of the principal block of Category $\mathcal{O}$ for any finite coxeter group. In this setting, we compute the extensions between Verma modules for dihedral groups.
We consider a general parabolic category $\mathcal{O}_{S}$ over symmetrizable Kac-Moody algebras. We introduce a reduction rule of hom-spaces between parabolic Verma modules over different Kac-Moody algebras which yield some applications on parabolic Verma module homomorphisms.
A practical method for constructing a nontrivial homomorphsim between Verma modules is described.
We defined the hyperelliptic Heisenberg algebra as the Heisenberg subalgebra of a hyperlliptic Krichever-Novikov algebra. Then we gave a explicit irreducibility criteria for $φ$-Verma modules for these algebras.
Let $\mathfrak{g}$ be a classial Lie algebra and $\mathfrak{p}$ be a maximal parabolic subalgebra. Let $M$ be a generalized Verma module induced from a one dimensional representation of $\mathfrak{p}$. Such $M$ is called a scalar type generalized Verma module. Its simple quotient $L$ is a highest weight moudle. In this paper, we will determine the reducibility of such scalar type generalized Verma modules by computing the Gelfand-Kirillov dimension of $L$.
Generalized Verma modules over sl(m+1) induced from simple highest weight modules
本文由单最高权 gl(m)-模构造了 sl(m+1) 上的广义 Verma 模，给出了其单性判据，并建立了该广义 Verma 模与张量模之间的等价关系。
In this paper we give a sum formula for the radical filtration of generalized Verma modules in any (possibly singular) blocks of parabolic BGG category which can be viewed as a generalization of Jantzen sum formula for Verma modules in the usual BGG category $\mathcal{O}$. Combined with Jantzen coefficients, we determine the radical filtrations for all basic generalized Verma modules. The proof makes use of the graded version of parabolic BGG category. Explicit formulae for the graded decomposition numbers and inverse graded decomposition numbers of generalized Verma modules in any (possibly s…
Comments on extensions of Verma modules in the Bernstein-Gelfand-Gelfand Category O.
Let $\mathfrak g(G,λ)$ denote the deformed generalized Heisenberg-Virasoro algebra related to a complex parameter $λ\neq-1$ and an additive subgroup $G$ of $\mathbb C$. For a total order on $G$ that is compatible with addition, a Verma module over $\mathfrak g(G,λ)$ is defined. In this paper, we completely determine the irreducibility of these Verma modules.
For a simple Lie superalgebra of type BDFG, we give explicit formulas for singular vectors in a Verma module of highest weight $λ- ρ$, which have weight $s_γλ- ρ$ for certain positive non-isotropic roots $γ.$ This implies the existence of a nonzero homomorphism between the corresponding Verma modules.
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。
We compute the homology of the complexes of finite Verma modules over the annihilation superalgebra $\mathcal A(K'_{4})$, associated with the conformal superalgebra $K'_{4}$, obtained in \cite{K4}. We use the computation of the homology in order to provide an explicit realization of all the irreducible quotients of finite Verma modules over $\mathcal A(K'_{4})$.
The main purpose of the paper is to establish new tools in the study of $\mathcal{O}^\mathfrak{p}$. We introduce the Jantzen coefficients of generalized Verma modules. It comes from the Jantzen's simplicity criteria for generalized Verma modules and has a deep relation with the structure of $\mathcal{O}^\mathfrak{p}$. We develop a reduction process to compute those coefficients. For which we need to consider generalized Verma modules induced from maximal parabolic subalgebras having maximal nontrivial singularity, so called basic generalized Verma modules. The classification of such modules is…
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。
In this paper we study the superalgebra $A_n$, introduced by the authors in previous work on categorification of Verma modules for quantum $\mathfrak{sl}_2$. The superalgebra $A_n$ is akin to the nilHecke algebra, and shares similar properties. In particular, we prove a uniqueness result about 2-Verma modules on $\Bbbk$-linear 2-categories.
We give a complete combinatorial classification of those parabolic Verma modules in the principal block of the parabolic category $\mathcal{O}$ associated to a minimal or a maximal parabolic subalgebra of the special linear Lie algebra for which the answer to Kostant's problem is positive.
We study non-standard Verma type modules over the Kac-Moody queer Lie superalgebra $\mathfrak{q}(n)^{(2)}$. We give a sufficient condition under which such modules are irreducible. We also give a classification of all irreducible diagonal $\mathbb{Z}$-graded modules over certain Heisenberg Lie superalgebras contained in $\mathfrak{q}(n)^{(2)}$.
We compute the homology of the first and third quadrants of the complexes of finite Verma modules over the annihilation superalgebra $\mathcal{A}(CK_{6})\cong E(1,6)$, associated with the conformal superalgebra $CK_6$, obtained in \cite{ck6}. This computation allows us to explicitly realize the irreducible quotients of degenerate finite Verma modules over $\mathcal{A}(CK_{6})$.
Graded extensions of Verma modules
本文研究 BGG 范畴 $\mathcal{O}$ 中分次 Verma 模的扩张，阐明 $R$-多项式系数所含信息，给出基于 Kazhdan–Lusztig 组合学的维数上界，并完全确定 $\mathfrak{sl}_4$ 正则块中的扩张及意外高次扩张。
Let $\preceq$ be a compatible total order on the additive group $\mathbb{Z}^2$, and $L$ be the rank two Heisenberg-Virasoro algebra. For any $\mathbf{c}=(c_1,c_2,c_3,c_4) \in \mathbb{C}^4$, we define $\mathbb{Z}^2$-graded Verma module $M(\mathbf{c}, \preceq)$ for the Lie algebra $L$. A necessary and sufficient condition for the Verma module $M(\mathbf{c}, \preceq)$ to be irreducible is provided. Moreover, the maximal $\mathbb{Z}^2$-graded submodules of the Verma module $M(\mathbf{c}, \preceq)$ are characterized when $M(\mathbf{c}, \preceq)$ is reducible.
An Intersection-Dimension Formula for Preprojective Modules of Type $\widetilde{D}_n$
本文证明了型 $\widetilde{D}_n$ 路代数预投射模的交集-维数公式，利用已知的 Auslander-Reiten 箭图结构直接证明了两条曲线交集数与所表示模的第一扩张空间维数相等的等式关系。
We study the homomorphisms between scalar generalized Verma modules. We conjecture that any homomorphism between is composition of elementary homomorphisms. The purpose of this article is to show the conjecture is affirmative for many parabolic subalgebras under the assumption that the infinitesimal characters are regular.
We shall give a uniform expression and a uniform calculation for the b-functions of prehomogeneous vector spaces of commutative parabolic type, which were previously calculated by case-by-case analysis. Our method is a generalization of Kashiwara's approach using the universal Verma modules. We shall also give a new proof for the criterion of the irreducibility of the generalized Verma module in terms of b-functions due to Suga, Gyoja, Wachi.
Classification of degenerate Verma modules over $E(4,4)$
本文对$E(4,4)$上退化Verma模块进行分类，完善了例外线性紧李超代数上Verma模块的描述，退化模与它们之间的映射产生无限双边复形，可视为de Rham复形的推广。
We consider imaginary Verma modules for quantum affine algebraU_q(\widehat{\mathfrak{sl}(2)}) and define a crystal-like base which we call an imaginary crystal basis using the Kashiwara algebra K_q constructed in earlier work of the authors. In particular, we prove the existence of imaginary like bases for a suitable category of reduced imaginary Verma modules for U_q(\widehat{\mathfrak{sl}(2)}).
We consider one of the most natural extended affine Lie lagebras, the algebra $sl_2({\mathbb C}_q)$ and begin a theory of its representations. In particular, we study a class of imaginary Verma modules, obtain a criterion of irreducibility and describe their submodule structure in "general position".
For a restricted Verma module of an affine Kac-Moody algebra at the critical level we describe the Jantzen filtration and give an alternating sum formula which corresponds to the Jantzen sum formula of a baby Verma module over a modular Lie algebra. This also implies a new proof of the linkage principle which was already deduced by Arakawa and Fiebig.
A recent result of N. Abe implies that the Gabber-Joseph conjecture is true for the first-degree extensions between Verma modules with regular integral highest weights.
$\mathfrak b_1$-Verma $\mathfrak b_2$-dual Verma supermodules
本研究针对I型基本经典李超代数上同时为两个不同Borel子代数对应的Verma模与对偶Verma模的模结构，通过分析有限杨格边收缩，证明其必同构于特异或反特异Borel子代数的Verma模，方法可推广至多数基本经典李超代数。
We describe the structure of a Verma module with a generic highest weight at the critical level over a symmetrizable affine Lie superalgebra not of the type A(2k,2l)^{(4)}. We obtain the character formula for a simple module with a generic highest weight at the critical level conjectured by V. G. Kac and D. A. Kazhdan.
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 develop a general technique of constructing new irreducible weight modules for any affine Kac-Moody algebra using the parabolic induction, in the case when the Levi factor of a parabolic subalgebra is infinite-dimensional and the central charge is nonzero. Our approach uniforms and generalizes all previously known results with imposed restrictions on inducing modules. We also define generalized Imaginary Wakimoto modules which provide an explicit realization for generic generalized Imaginary Verma modules.
We introduce the notion of essential support of a simple Gelfand-Tsetlin $\mathfrak{gl}_n$-module as an important tool towards understanding the character formula of such module. This support detects the weights in the module having maximal possible Gelfand-Tsetlin multiplicities. Using combinatorial tools we describe the essential supports of the simple socles of the universal tableaux modules. We also prove that every simple Verma module appears as a socle of a universal tableaux module and hence obtain a description of the essential supports of all simple Verma modules. As a consequence, we…
We introduce (n+1)-preprojective algebras of algebras of global dimension n. We show that if an algebra is n-representation-finite then its (n+1)-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)-preprojective algebra is (n+1)-Calabi-Yau, and, more precisely, it is the (n+1)-Amiot cluster category of the stable n-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)-cluster tilting object. We show that even if the (n+1)-preprojective algebra is not self-injective, under certain assumptions …
It is shown by Barchini, Kable, and Zierau that conformally invariant systems of differential operators yield explicit homomorphisms between certain generalized Verma modules. In this paper we determine whether or not the homomorphisms arising from such systems of first and second order differential operators associated to maximal parabolic subalgebras of quasi-Heisenberg type are standard.
We classify contravariant pairings between standard Whittaker modules and Verma modules over a complex semisimple Lie algebra. These contravariant pairings are useful in extending several classical techniques for category $\mathcal{O}$ to the Miličić--Soergel category $\mathcal{N}$. We introduce a class of costandard modules which generalize dual Verma modules, and describe canonical maps from standard to costandard modules in terms of contravariant pairings. We show that costandard modules have unique irreducible submodules and share the same composition factors as the corresponding standard …
Suppose that $Q$ is a finite quiver and $G\subseteq \Aut(Q)$ is a finite group, $k$ is an algebraic closed field whose characteristic does not divide the order of $G$. For any algebra $Λ=kQ/{\mathcal {I}}$, $\mathcal {I}$ is an arbitrary ideal of path algebra $kQ$, we give all the indecomposable $ΛG$-modules from indecomposable $Λ$-modules when $G$ is abelian. In particular, we apply this result to the deformed preprojective algebra $Π_{Q}^λ$, and get a reflection functor for the module category of $Π_{Q}^λG$. Furthermore, we construct a new quiver $Q_{G}$ and prove that $Π_{Q}^λG$ is Morita e…
We study the restricted Verma modules of an affine Kac-Moody algebra at the critical level with special emphasis on their Jordan-H"older multiplicities. The Feigin-Frenkel conjecture gives a formula for these multiplicities that involves the periodic Kazhdan-Lusztig polynomials. We prove this conjecture for all subgeneric blocks and for the case of anti-dominant simple subquotients.
The Category of reduced imaginary Verma modules
本文研究任意仿射李代数在自然Borel子代数和零中心荷条件下的范畴O类似物，证明其半单性并以约化虚Verma模为单对象，推广了仿射sl(2)的已有结果。
Assuming the Lusztig conjecture on the irreducible characters for reductive algebraic groups in positive characteristic $p$, which is now a theorem for large $p$, we show that the modules for their Frobenius kernels induced from the simple modules of $p$-regular highest weights for their parabolic subgroups are rigid and determine their Loewy series.
With this paper we start the study of reducible representations of the Jacobi algebra with the ultimate goal of constructing differential operators invariant w.r.t. the Jacobi algebra. In this first paper we show examples of the low level singular vectors of Verma modules over the Jacobi algebra. According to our methodology these will produce the invariant differential operators.
Let $Q$ be a wild $n$-Kronecker quiver, i.e., a quiver with two vertices, labeled by 1 and 2, and $n\geq 3$ arrows from 2 to 1. The indecomposable regular modules with preprojective Gabriel-Roiter submodules, in particular, those $τ^{-i}X$ with $\udim X=(1,c)$ for $i\geq 0$ and some $1\leq c\leq n-1$ will be studied. It will be shown that for each $i\geq 0$ the irreducible monomorphisms starting with $τ^{-i}X$ give rise to a sequence of Gabriel-Roiter inclusions, and moreover, the Gabriel-Roiter measures of those produce a sequence of direct successors. In particular, there are infinitely many…
Branching problem of tensoring two Verma modules and its application to differential symmetry breaking operators
本文研究半单李代数中两个Verma模张量积的分支问题，为Kobayashi-Pevzner发现的sl2情形重数一失效参数提供新刻画，并描述不可分解分量及同构条件。
Given a Lie superalgebra $\frak g$ with a subalgebra $\frak g_{\geq 0}$, and a finite-dimensional irreducible $\frak g_{\geq 0}$-module $F$, the induced $\frak g$-module $M(F)=U({\frak g}) \otimes_{U(\frak g_{\geq 0})} F $ is called a finite Verma module. In the present paper we classify the non-irreducible finite Verma modules over the largest exceptional linearly compact Lie superalgebra ${\frak g}=E(5,10)$ with the subalgebra $\frak g_{\geq 0}$ of minimal codimension. This is done via classification of all singular vectors in the modules $M(F)$. Besides known singular vectors of degree 1,2,…
This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type $A$ we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type $A$, which finally determines the so-called rough structure of generalized Verma modules. On the way we present several categorification results and give the positive answer to Kostant's problem from \cite{Jo} in many cases.…
Preprojective categories of type A
本文研究了类型A的连续预射影代数，定义了优雅模和排列理想，并展示了其与τ-刚性的关系。
In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $G_{2(2)}$ which is split real form of $G_2$. We give the classification of reducible Verma modules $G_2$. We give also the singular vectors between these modules, thus setting the stage for construction of the invariant differential operators over $G_{2(2)}$.
This paper studies connections between the preprojective modules over the path algebra of a finite connected quiver without oriented cycles, the (+)-admissible sequences of vertices, and the Weyl group. For each preprojective module, there exists a unique up to a certain equivalence shortest (+)-admissible sequence annihilating the module. A (+)-admissible sequence is the shortest sequence annihilating some preprojective module if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. These statements have the following app…
In this paper, we give sufficient properties for a finite dimensional graded algebra to be a higher preprojective algebra. These properties are of homological nature, they use Gorensteiness and bimodule isomorphisms in the stable category of Cohen-Macaulay modules. We prove that these properties are also necessary for $3$-preprojective algebras using \cite{Kel11} and for preprojective algebras of higher representation finite algebras using \cite{Dugas}.
Let $G$ be a reductive algebraic group over an algebraically closed field of positive characteristic, $G_1$ the Frobenius kernel of $G$, and $T$ a maximal torus of $G$. We show that the $G_1T$-Verma modules of singular highest weights are all rigid, determine their Loewy length, and describe their Loewy structure using the periodic Kazhdan-Lusztig $Q$-polynomials. We assume that the characteristic of the field is large enough that, in particular, Lusztig's conjecture for the irreducible $G_1T$-characters hold.
In this paper we close the cases that were left open in our earlier works on the study of conformally invariant systems of second-order differential operators for degenerate principal series. More precisely, for these cases, we find the special values of the systems of differential operators, and determine the standardness of the homomorphisms between the generalized Verma modules, that come from the conformally invariant systems.
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。
Symplectic reflection algebras arise in many different mathematical disciplines: integrable systems, Lie theory, representation theory, differential operators, symplectic geometry. In this paper, we introduce baby Verma modules for symplectic reflection algebras of complex reflection groups at parameter t=0 (the so--called rational Cherednik algebras at parameter t=0) and present their most basic properties. By analogy with the representation theory of reductive Lie algebras in positive characteristic, we believe these modules are fundamental to the understanding of the representation theory a…
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