Julia Pevtsova, Jim Stark
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.
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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 nilpotent cone of a reductive Lie algebra has a desingularization given by thecotangent bundle of the flag variety. Analogously, the nullcone of a cartesianpower of the algebra has a desingularization given by a vector bundle over theflag variety. As for the nullcone, the subvariety of elements whose componentsare in a same Borel subalgebra, has a desingularization given by a vector bundle overthe flag variety. In this note, some properties of these varieties are given. Forthe study of the commuting variety, the analogous variety to the flag variety isthe closure in the Grassmannian of the…
Let G be a connected simple algebraic group over an algebraically closed field k of characteristic p > 0, and g := Lie(G). We additionally assume that G is standard and is of type An. Motivated by the investigation of the geometric properties of the varieties E(r, g) of r-dimensional elementary subalgebras of a restricted Lie algebra g, we will show in this article the irreducible components of E(rkp(g)-1,g) when rkp(g) is the maximal dimension of an elementary subalgebra of g.
This is a continuation of the sequence of papers \cite{HN2}, \cite{H99} in the study of the cocenters and class polynomials of affine Hecke algebras $\ch$ and their relation to affine Deligne-Lusztig varieties. Let $w$ be a $P$-alcove element, as introduced in \cite{GHKR} and \cite{GHN}. In this paper, we study the image of $T_w$ in the cocenter of $\ch$. In the process, we obtain a Bernstein presentation of the cocenter of $\ch$. We also obtain a comparison theorem among the class polynomials of $\ch$ and of its parabolic subalgebras, which is analogous to the Hodge-Newton decomposition theor…
The purpose of this paper is to bring together various loose ends in the theory of integrable systems. For a semisimple Lie algebra $\mathfrak g$, we obtain several results on completeness of homogeneous Poisson-commutative subalgebras of ${\mathcal S}(\mathfrak g)$ on coadjoint orbits. This concerns, in particular, Mishchenko-Fomenko and Gelfand-Tsetlin subalgebras.
Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $\mathfrak p$ be the Lie algebra of $P$. We consider the commuting variety $\mathcal C(\mathfrak p) = \{(X,Y) \in \mathfrak p \times \mathfrak p \mid [X,Y] = 0\}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $\mathcal C(\mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $\mathfrak p$. As a conseque…
Varieties of subalgebras and endotrivial modules
本文结合两种方法，描述了特征 p≥3 的完美域上超可解受限李代数的内平凡模。
Let $G$ be a semisimple, simply connected, algebraic group over an algebraically closed field $k$ with Lie algebra $\frak g$. We study the spaces of parahoric subalgebras of a given type containing a fixed nil-elliptic element of $\frak g\otimes k((π))$, i.e. fixed point varieties on affine flag manifolds. We define a natural class of $k^*$-actions on affine flag manifolds, generalizing actions introduced by Lusztig and Smelt. We formulate a condition on a pair $(N,f)$ consisting of $N\in\frak{g}\otimes k((π))$ and a $k^*$-action $f$ of the specified type which guarantees that $f$ induces an a…
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $\mathbbm k$ of characteristic zero. We consider the commuting variety $\mathcal C(\mathfrak u)$ of the nilradical $\mathfrak u$ of the Lie algebra $\mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $\mathfrak u$ with only a finite number of orbits, we verify that $\mathcal C(\mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {\em distinguished} $B$-orbits in $\mathfrak u$. We observe that in general $\mathcal C(\mathfrak u)$ is not equidim…
A finite-dimensional Lie algebra $L$ over a field $F$ of characteristic zero is called elementary if each of its subalgebras has trivial Frattini ideal; it is an $A$-algebra if every nilpotent subalgebra is abelian. This paper is a continuation of the study of these algebras initiated by the authors in `Elementary Lie Algebras and Lie A-algebras', J. Algebra 312 (2007), 891--901. If we denote by $\mathcal{A}$, $\mathcal{G}$, $\mathcal{E}$, $\mathcal{L}$, $Φ$ the classes of $A$-algebras, almost algebraic algebras, $E$-algebras, elementary algebras and $ϕ$-free algebras respectively, then it is …
Birational equivalence of the Zassenhaus varieties for basic classical Lie superalgebras and their purely-even reductive Lie subalgebras in odd characteristic
本文研究奇特征代数闭域上基本经典李超代数的泛包络代数中心结构，证明其分式域与纯偶约化李子代数泛包络代数中心的分式域同构，进而得到两类Zassenhaus簇的双有理等价性及中心谱的有理性。
For $G$ a connected, reductive group over an algebraically closed field $k$ of large characteristic, we use the canonical Springer isomorphism between the nilpotent variety of $\mathfrak{g}:=\mathrm{Lie}(G)$ and the unipotent variety of $G$ to study the projective variety of elementary subalgebras of rank $r$, denoted $\mathbb{E}(r,\mathfrak{g})$. In the case that $G$ is defined over $\mathbb{F}_p$, we define the category of $\mathbb{F}_p$-expressible subalgebras of $\mathfrak{g}$, and prove that this category is isomorphic to Quillen's category of elementary abelian subgroups of the finite Ch…
In a previous paper we constructed rank and support variety theories for "quantum elementary abelian groups," that is, tensor products of copies of Taft algebras. In this paper we use both variety theories to classify the thick tensor ideals in the stable module category, and to prove a tensor product property for the support varieties.
We initiate the study of representations of elementary abelian $p$-groups via restrictions to truncated polynomial subalgebras of the group algebra generated by $r$ nilpotent elements, $k[t_1,..., t_r]/(t^p_1,..., t_r^p)$. We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several general constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic…
Quantum Grassmannians and their Associated Quantum Schubert Varieties at roots of unity
本文研究了单位根处量子代数的 PI degree，通过证明相关矩阵不变因子为 2 的幂来显式计算分划子代数的 PI degree，并利用非交换去齐次化方法推导出量子舒伯特簇和量子格拉斯曼流形的 PI degree。
In \cite{BKN} the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties for Kac supermodules for Type I Lie superalgebras and the simple supermodules for $\mathfrak{gl}(m|n)$. T…
We develop a support theory for elementary supergroup schemes, over a field of positive characteristic $p\ge 3$, starting with a definition of a $π$-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and $π$-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra $k[t,τ]/(t^p-τ^2)$, where $t$ has even degree and $τ$ has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup…
We initiate the investigation of the projective varieties $\mathbb E(r,\mathfrak g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $\mathfrak g$ for various $r \geq 1$. These varieties $\mathbb E(r,\mathfrak g)$ are the natural ambient varieties for generalized support varieties for restricted representations of $\mathfrak g$. We identify these varieties in special cases, revealing their interesting and varied geometric structures. We also introduce invariants for a finite dimensional $\mathfrak u(\mathfrak g)$-module $M$, the local $(r,j)$-radical rank and local…
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…
Let $(\mathfrak{g},[p])$ be a restricted Lie algebra over an algebraically closed field $k$ of characteristic $p\!\ge \!3$. Motivated by the behavior of geometric invariants of the so-called $(\mathfrak{g},[p])$-modules of constant $j$-rank ($j \in \{1,\ldots,p\!-\!1\}$), we study the projective variety $\mathbb{E}(2,\mathfrak{g})$ of two-dimensional elementary abelian subalgebras. If $p\!\ge\!5$, then the topological space $\mathbb{E}(2,\mathfrak{g}/C(\mathfrak{g}))$, associated to the factor algebra of $\mathfrak{g}$ by its center $C(\mathfrak{g})$, is shown to be connected. We give applicat…
In this paper we use invariant theory to develop the notion of cohomological detection for Type I classical Lie superalgebras. In particular we show that the cohomology with coefficients in an arbitrary module can be detected on smaller subalgebras. These results are used later to affirmatively answer questions, which were originally posed in \cite{BKN1} and \cite{BaKN}, about realizing support varieties for Lie superalgebras via rank varieties constructed for the smaller detecting subalgebras.
We define a rank variety for a module of a noncocommutative Hopf algebra $A = Λ\rtimes G$ where $Λ= k[X_1, ..., X_m]/(X_1^{\ell}, ..., X_m^{\ell})$, $G = ({\mathbb Z}/\ell{\mathbb Z})^m$, and $\text{char} k$ does not divide $\ell$, in terms of certain subalgebras of $A$ playing the role of "cyclic shifted subgroups". We show that the rank variety of a finitely generated module $M$ is homeomorphic to the support variety of $M$ defined in terms of the action of the cohomology algebra of $A$. As an application we derive a theory of rank varieties for the algebra $Λ$. When $\ell=2$, rank varieties…
We prove that quantized multiplicative quiver varieties and quantum character varieties define sheaves of Azumaya algebras over the corresponding classical moduli spaces, and we prove that the Azumaya locus of the Kauffman bracket skein algebras contains the smooth locus, proving a strong form of the Unicity Conjecture of Bonahon and Wong. The proofs exploit a strong compatibility between quantum Hamiltonian reduction and the quantum Frobenius homomorphism as it arises in each setting. We therefore introduce the concepts of Frobenius quantum moment maps and their Hamiltonian reduction, and of …
We provide a Lagrangian construction for the fixed-point subalgebra, together with its idempotent form, in a quasi-split symmetric pair of type $A_{n-1}$. This is obtained inside the limit of a projective system of Borel-Moore homologies of the Steinberg varieties of $n$-step isotropic flag varieties. Arising from the construction are a basis of homological origin for the idempotent form and a geometric realization of rational modules.
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 prove that, if $A$ is a positively graded, graded commutative, local, finite Hopf algebra, its cohomology is finitely generated, thus unifying classical results of Wilkerson and Hopkins-Smith, and of Friedlander-Suslin. We do this by showing the existence of conormal elementary quotients.
Support varieties for Lie superalgebras over the complex numbers were introduced in \cite{BKN1} using the relative cohomology. In this paper we discuss finite generation of the relative cohomology rings for Lie superalgebras, we formulate a definition for subalgebras which detect the cohomology, also discuss realizability of support varieties. In the last section as an application we compute the relative cohomology ring of the Lie superalgebra $\overline{S}(n)$ relative to the graded zero component $\overline{S}(n)_0$ and show that this ring is finitely generated. We also compute support varie…
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.
Starting from a split semisimple real Lie group G with trivial center, we define a family of varieties with additional structures. We describe them as the cluster X-varieties, as defined in math.AG/0311245. In particular they are Poisson varieties. We define Poisson maps of them to the group G with the standard Poisson-Lie structure. We introduce an operation of amalgamation of cluster varieties. Our varieties are amalgamations of elementary ones, assigned to positive simple roots of the root system of G. Some of them are very closely related to the double Bruhat cells. This paper is a buildin…
Let $(W, I)$ be a finite Coxeter group. In the case where $W$ is a Weyl group, Berenstein and Kazhdan in \cite{BK} constructed a monoid structure on the set of all subsets of $I$ using unipotent $χ$-linear bicrystals. In this paper, we will generalize this result to all types of finite Coxeter groups (including non-crystallographic types). Our approach is more elementary, based on some combinatorics of Coxeter groups. Moreover, we will calculate this monoid structure explicitly for each type.
Affine flag varieties and quantum symmetric pairs
本文通过仿射C型部分旗簇的Schur与Lusztig代数，构造了仿射型量子对称对的几何实现，并建立了相应量子代数的典范基及其正性性质。
In this paper, we deal with the $\mathcal{U}(\mathfrak{g})$-action on a $\mathfrak{g}$-module on which a larger algebra $\mathcal{A}$ acts irreducibly. Under a mild condition, we will show that the support of the $\mathcal{Z}(\mathfrak{g})$-action is a union of affine subspaces in the dual of a Cartan subalgebra modulo the Weyl group action. As a consequence, we propose a definition of a Cartan subalgebra for such a $\mathfrak{g}$-module. The support of the $\mathcal{Z}(\mathfrak{g})$-module is an algebraic counterpart of the support of the measure in the irreducible decomposition of a unitary…
Let $u_ζ(g)$ denote the small quantum group associated to the simple complex Lie algebra $g$, with parameter $q$ specialized to a primitive $\ell$-th root of unity $ζ$ in the field $k$. Generalizing a result of Cline, Parshall and Scott, we show that if $M$ is a finite-dimensional $u_ζ(g)$-module admitting a compatible torus action, then the injectivity of $M$ as a module for $u_ζ(g)$ can be detected by the restriction of $M$ to certain root subalgebras of $u_ζ(g)$. If the characteristic of $k$ is positive, then this injectivity criterion also holds for the higher Frobenius--Lusztig kernels $U…
Let $\frak g$ be a reductive Lie algebra over an algebraically closed field of characteristic 0 and $\frak k$ be a reductive in $\frak g$-subalgebra. Let $M$ be a finitely generated (possibly, infinite-dimensional) $\frak g$-module. We say that $M$ is a $(\frak g, \frak k)$-module if $M$ is a direct sum of a (possibly, infinite) amount of simple finite-dimensional $\frak k$-modules. We say that $M$ is of finite type if $M$ is a $(\frak g, \frak k)$-module and Hom$_\frak k(V, M)<\infty$ for any simple $\frak k$-module $V$. Let $X$ be a variety of all Borel subalgebras of $\frak g$. Let $M$ be a…
Support varieties for any finite dimensional algebra over a field were introduced by Snashall-Solberg using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected…
We construct rank varieties for the Drinfel'd double of the Taft algebra and for U_q(sl2). For the Drinfel'd double when n=2 this uses a result which identifies a family of subalgebras that control projectivity of A-modules whenever A is a Hopf algebra satisfying a certain homological condition. In this case we show that our rank variety is homeomorphic to the cohomological support variety. We also show that Ext^*(M,M) is finitely generated over the cohomology ring of the Drinfel'd double for any finitely-generated module M.
Let $\mathcal{O}$ be a Richardson nilpotent orbit in a simple Lie algebra $\mathfrak{g}$ over $\mathbb C$, induced from a Levi subalgebra whose simple roots are orthogonal short roots. The main result of the paper is a description of a minimal set of generators of the ideal defining $\overline{ \mathcal{O}}$ in $S \mathfrak{g}^*$. In such cases, the ideal is generated by bases of at most two copies of the representation whose highest weight is the dominant short root, along with some fundamental invariants. This extends Broer's result for the subregular nilpotent orbit. Along the way we give a…
We introduce the notion of a generalized spin representation of the maximal compact subalgebra of a symmetrizable Kac-Moody algebra in order to show that, if defined over a formally real field, every such subalgebra has a non-trivial reductive finite-dimensional quotient. The appendix illustrates how to compute the isomorphism types of these quotients for the real $E_n$ series. In passing this provides an elementary way of determining the isomorphism types of the maximal compact subalgebras of the semisimple split real Lie algebras of types $E_6$, $E_7$, $E_8$.
Affine flag varieties of type D
本文建立仿射D型完全与部分旗簇的格表示，确定其完全旗簇对应的卷积代数同构于扩张仿射Hecke代数，证明该代数存在单项式基与典范基并给出乘法正性性质，拓展了仿射旗簇的几何表示理论。
The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible components of Springer fibers. In type A, a construction due to Richardson implies that every nilpotent orbit contains at least one smooth orbital variety and every Springer fiber contains at least one smooth component. In this paper, we show that this property is also true for the other classical cases. Our proof uses the interpretation of Springer fibers …
Geometric Approach to I-Quantum Group of Affine Type D
本文采用几何方法研究仿射型D相关的Schur代数与Lusztig代数结构，证明Lusztig代数的一个子代数经稳定化得到仿射$\mathfrak{sl}$型量子群的余理想子代数，还构造了幂等量子代数的单项式基与典范基并确立其正性性质。
DAHAs of Type $C^\vee C_n$ and Character Varieties
本文研究$C^\vee C_n$型双仿射Hecke代数的球面子代数，在$q=1$经典水平建立其与四穿孔黎曼球面特征簇的关联，证明了一个已有猜想，还得到三角van Diejen系统的完备相空间并显式积分其动力学、给出哈密顿约化构造方式。
Let $\mathfrak{g}$ be the $p$-dimensional Witt algebra over an algebraically closed field $k$ of characteristic $p>3$. Let $\mathscr{N}={x\in\ggg\mid x^{[p]}=0}$ be the nilpotent variety of $\mathfrak{g}$, and $\mathscr{C}(\mathscr{N}):=\{(x,y)\in \mathscr{N}\times\mathscr{N}\mid [x,y]=0\}$ the nilpotent commuting variety of $\mathfrak{g}$. As an analogue of Premet's result in the case of classical Lie algebras [A. Premet, Nilpotent commuting varieties of reductive Lie algebras. Invent. Math., 154, 653-683, 2003.], we show that the variety $\mathscr{C}(\mathscr{N})$ is reducible and equidimens…
It is shown that any generalized Kac-Moody Lie algebra g that has no mutually orthogonal imaginary simple roots can be written as the vector space direct sum of a Kac-Moody subalgebra and subalgebras isomorphic to free Lie algebras over certain modules for the Kac-Moody subalgebra. Also included is a detailed discussion of Borcherds' construction of the Monster Lie algebra from a vertex algebra and an elementary proof of Borcherds' theorem relating Lie algebras with `an almost positive definite bilinear form' to generalized Kac-Moody algebras. (Preprint version 1996)
Small modules with interesting rank varieties
本文研究初等阿贝尔p-群上群代数模的秩簇，给出不可分解模Green顶点包含该群的充分条件，并探索寻找指定秩簇的小模的问题，还考察了对称群的特定单模。
Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras
本文研究A型分裂实Kac-Moody代数极大紧子代数的有限维旋量表示，分析其可约性、半单性等性质，借助旋量扩张Weyl群得到表示矩阵由实根的部分参数化结果。
$p$-adic Maass--Shimura operators on $μ$-ordinary Igusa varieties
该研究针对霍奇型志村簇，将p进整近全纯自守形式上的代数马阿斯-志村微分算子扩展到μ-普通曼托万伊古萨簇，通过p进傅里叶理论等方法得到相关代数作用，扩展、澄清并恢复了先前构造。
The purpose of this thesis is to present certain viewpoints on the geometric representation theory of Nakajima cyclic quiver varieties, in relation to the Maulik-Okounkov stable basis. Our main technical tool is the shuffle algebra, which arises as the K-theoretic Hall algebra of the double cyclic quiver. We prove the isomorphism between the shuffle algebra and the quantum toroidal algebra U_{q,t}(sl_n^^), and identify the quotients of Verma modules for the shuffle algebra with the K-theory groups of Nakajima cyclic quiver varieties, which were studied by Nakajima and Varagnolo-Vasserot. The s…
We define slope subalgebras in the shuffle algebra associated to a (doubled) quiver, thus yielding a factorization of the universal R-matrix of the double of the shuffle algebra in question. We conjecture that this factorization matches the one defined by [1, 18, 32, 33, 34] using Nakajima quiver varieties.
Bethe subalgebras in Yangians and Kirillov-Reshetikhin crystals
本文在A型Yangian的Bethe子代数谱上构造仿射晶体，通过退化到万有非齐次Gaudin子代数并联系Feigin-Frenkel中心，证明其与Kirillov-Reshetikhin晶体一致，进而证明相关单值性猜想。
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