Dave Witte
We prove that each real semisimple Lie algebra G has a Q-form, such that every real representation of G can be realized over the rational numbers Q. This was previously proved by M.S.Raghunathan (and rediscovered by P.Eberlein) in the special case where G is compact.
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
This paper is a continuation of the theory of cyclic elements in semisimple Lie algebras, developed by Elashvili, Kac and Vinberg. Its main result is the classification of semisimple cyclic elements in semisimple Lie algebras. The importance of this classification stems from the fact that each such element gives rise to an integrable hierarchy of Hamiltonian PDE of Drinfeld-Sokolov type.
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 determine when there exists a nonzero homomorphism between principal series representations of a complex semisimple Lie group. We also determines the existence of homomorphisms between twisted Verma modules.
We study rewriting properties of the column presentation of plactic monoid for any semisimple Lie algebra such as termination and confluence. Littelmann described this presentation using L-S paths generators. Thanks to the shapes of tableaux, we show that this presentation is finite and convergent. We obtain as a corollary that plactic monoids for any semisimple Lie algebra satisfy homological finiteness properties.
Regular extreme semisimple Lie algebras
本文证明了所有单 Lie 代数均为正则极点，且无非单的半单 Lie 代数是正则极点，推广了 $A_n$ 型的结果。
We suggest a method of constructing special nonunitary representations of semisimple Lie groups using representations of Iwasawa subgroups. As a typical example, we study the group $U(2,2)$.
The main goal of this paper is to prove the following theorem: Let $\frak k$ be an $\frak {sl}_2$-subalgebra of a semisimple Lie algebra $\frak g$, none of whose simple factors is of type $A1$. Then there exists a positive integer $b(\frak k, \frak g)$, such that for every irreducible finite dimensional $\frak g$-module $V$, there exists an injection of $\frak k$-modules $W \to V$, where $W$ is an irreducible $\frak k$-module of dimension less than $b(\frak k, \frak g)$. This result was announced in math.RT/0310140.
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…
We describe all locally semisimple subalgebras and all maximal subalgebras of the finitary Lie algebras $\gl(\infty), \sl(\infty), \so(\infty)$, and $\sp(\infty)$. For simple finite--dimensional Lie algebras these classes of subalgebras have been described in the classical works of A. Malcev and E. Dynkin. Key words (2000 MSC): 17B05 and 17B65.
Symmetric Biderivations on Complex Semisimple Lie Algebras
本文证明了有限维复半单李代数到其有限维模上的对称双导子均为平凡的，并据此分类了 Takiff 代数、辛振子代数及若干经典单李超代数上的双导子。
In this article, we give a simple explicit construction of an affine slice for the coadjoint action of a certain class of biparabolic (also called seaweed) subalgebras of a semisimple Lie algebra over an algebraically closed field of characteristic zero. In particular, this class includes all Borel subalgebras.
We consider the notion of a confluent spherical function on a connected semisimple Lie group, $G,$ with finite center and of real rank $1,$ and discuss the properties and relationship of its algebra with the well-known Schwartz algebra of spherical functions on $G.$
We construct a convenient basis for all real semisimple Lie algebras by means of an adapted Chevalley basis of the complexification. It determines rational and in fact half-integer structure constants which we express only in terms of the root system and the involution that defines the real structure.
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.
A $G$-grading on an algebra is called multiplicity free if each homogeneous component of the grading is 1-dimensional, where $G$ is an abelian group. We introduce skew root systems of Lie type and skew root systems of Jordan type respectively, and use them to construct multiplicity free gradings on semisimple Lie algebras and on semisimple Jordan algebras respectively. Under certain conditions the corresponding Lie (resp. Jordan) algebras are simple. Three families of skew root systems of Lie type (resp. of Jordan type) are constructed and the corresponding Lie (resp. Jordan) algebras are iden…
The classical Erd{\" o}s-Turan Inequality bounds how far a sequence of points in the circle is from being equidistributed in terms of its exponential moments. We prove an analogous inequality for all compact simply-connected semisimple Lie groups, bounding how far a sequence is from being equidistributed in the conjugacy classes of the group in terms of the moments of irreducible characters.
We give the exact contributions of Harish-Chandra transform, $(\mathcal{H}f)(λ),$ of Schwartz functions $f$ to the harmonic analysis of spherical convolutions and the corresponding $L^{p}-$ Schwartz algebras on a connected semisimple Lie group $G$ (with finite center). One of our major results gives the proof of how the Trombi-Varadarajan Theorem enters into the spherical convolution transform of $L^{p}-$ Schwartz functions.
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.
In this paper we develop algorithms of finding homogeneous spaces of semisimple non-compact Lie groups which do not admit compact Clifford-Klein forms. We propose a computer program which checks if the given homogeneous space has a non-vanishing cohomological obstruction (found by Tholozan) to compact Clifford-Klein forms. By a numerical experiment we show that there is a large class of homogeneous spaces satisfying Tholozan's condition.
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…
We consider spherical principal series representations of the semisimple Lie group of rank one $G=SO(n, 1; \mathbb K)$, $\mathbb K=\br, \bc, \bh$. There is a family of unitarizable representations $π_ν$ of $G$ for $ν$ in an interval on $\mathbb R^+$, the so-called complementary series, and subquotient or subrepresentations of $G$ for $ν$ being negative integers. We consider the restriction of $(π_ν, G)$ under the subgroup $H=SO(n-1, 1; \mathbb K)$. We prove the appearing of discrete components. The corresponding results for the exceptional Lie group $F_{4(-20)}$ and its subgroup $Spin(8,1)$ ar…
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。
We construct combinatorial (involutory) Gelfand models for the following diagram algebras in the case when they are semi-simple: Brauer algebra, its partial analogue, walled Brauer algebra, its partial analogue, Temperley-Lieb algebra, its partial analogue, walled Temperley-Lieb algebra, its partial analogue, partition algebra and its Temperley-Lieb analogue.
The Lie algbera of a compact semisimple Lie group G is determined by the degrees of the irreducible representations of G. However, two different groups can have the same representation degrees.
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。
These are lecture notes for a one semester introductory course I gave at Indiana University. The goal was to make this exposition as clear and elementary as possible. A particular emphasis is given on examples involving SU(1,1). These notes are in part based on lectures given by my graduate advisor Wilfried Schmid at Harvard University and PQR2003 Euroschool in Brussels as well as other sources.
In this expository paper we describe the theory of Harish-Chandra highest weight representations and their explicit geometric realizations.
We give a short elementary proof of the fact that connected semisimple real Lie groups satisfy property RD. The proof is based on a process of linearization.
We consider the spherical complementary series of rank one Lie groups $H_n=\SO_0(n, 1; \mathbb F)$ for $\mathbb F=\mathbb R, \mathbb C, \mathbb H$. We prove that there exist finitely many discrete components in its restriction under the subgroup $H_{n-1}=\SO_0(n-1, 1; \mathbb F)$. This is proved by imbedding the complementary series into analytic continuation of holomorphic discrete series of $G_n=SU(n, 1)$, $SU(n, 1)\times SU(n, 1)$ and $SU(2n, 2)$ and by the branching of holomorphic representations under the corresponding subgroup $G_{n-1}$.
In this short note, we give a description of semisimple orbits in the restricted Cartan type Lie algebras $W, S, H$.
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.
We prove that the canonical dimension of an admissible Banach space or a locally analytic representation of an arbitrary semisimple p-adic Lie group is either zero or at least half the dimension of a non-zero coadjoint orbit. This extends the results of Ardakov-Wadsley and Schmidt in the split semisimple case.
In this note we propose a method to classify homogeneous nilpotent elements in a real $Z_m$-graded semisimple Lie algebra $g$. Using this we describe the set of orbits of homogeneous elements in a real $Z_2$-graded semisimple Lie algebra. A classification of 4-vectors (resp. 4-forms) on $R^8$ can be given using this method.
In this note, we formulate an observation that "almost all" irreducible ordinary characters of finite groups of Lie type remain irreducible when restricted to the derived subgroups. To see this, key ingredients are some asymptotic results for conjugacy classes of finite groups of Lie type and strongly regular semisimple elements in dual groups.
Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.
Representations of Lie-Yamaguti algebras with semisimple enveloping Lie algebras
本文研究了包络李代数为半单的Lie-Yamaguti代数的表示问题，利用其包络李代数的表示及附加数据刻画了原代数的表示，并推广至无穷小$s$-流形情形。
We formulate and prove that there are "abundant" in nilpotent orbits in real semisimple Lie algebras, in the following sense. If S denotes the collection of hyperbolic elements corresponding the weighted Dynkin diagrams coming from nilpotent orbits, then S span the maximally expected space, namely, the (-1)-eigenspace of the longest Weyl group element. The result is used to the study of fundamental groups of non-Riemannian locally symmetric spaces.
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 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.
Subnormalisers of semisimple elements in finite groups of Lie type
本文研究了李型有限群中半单素数幂阶元素的子正规化子，通过确定Sylow tori正规化子的最大超群，推动了特征对应猜想和有限置换群中拟半正则元素存在的问题的研究。
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}…
We show the local rigidity of complex hyperbolic lattices in classical Hermitian semisimple Lie groups, $SU(np,p), Sp(2n+2,\mathbb R), SO^*(2n+2), SO(2n,2)$. This reproves or generalizes some results in \cite{GM, KKP, Klingler-inv, Pozzetti}.
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 $\mathfrak{g}$.
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.
Quantisation of semisimple real Lie groups
本文利用 Letzter 的量子对称对理论构造实半单李代数的量子化泛包络 *-代数，并通过“积分”实现任意半单代数实李群的群 C*-代数量子化。
We classify all pairs (m,e), where m is a positive integer and e is a nilpotent element of a semisimple Lie algebra, which arise in the classification of simple rational W-algebras.
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
Quantized semisimple Lie groups
本文从酉表示理论角度介绍了半单李群的q-形变，通过聚焦$\mathrm{sl}(2,\mathbb{C})$及其相关李群，探讨了量子群、量子化代数及Plancherel公式量子类似物等核心主题。
Let g be a finite dimensional complex semisimple Lie algebra, and let V be a finite dimensional represenation of g. We give a closed formula for the mth Frobenius-Schur indicator, m>1, of V in representation-theoretic terms. We deduce that the indicators take integer values, and that for a large enough m, the mth indicator of V equals the dimension of the zero weight space of V. For the classical Lie algebras sl(n), so(2n), so(2n+1) and sp(2n), this is the case for m greater or equal to 2n-1, 4n-5, 4n-3 and 2n+1, respectively.
Let $G$ be a complex reductive algebraic group, $g$ its Lie algebra and $h$ a reductive subalgebra of $g$, $n$ a positive integer. Consider the diagonal actions $G:g^n, N_G(h):h^n$. We study a relation between the algebra $C[h^n]^{N_G(h)}$ and its subalgebra consisting of restrictions to $h^n$ of elements of $C[g^n]^G$.
We prove that any coadjoint orbit with real eigenvalues of a complex semisimple Lie group, equipped with the real part of the canonical holomorphic symplectic form, is symplectomorphic to the cotangent bundle of a (partial) flag manifold. Moreover, we generalize this result to hyperbolic orbits in a real semisimple Lie algebra.
We consider the equivariant K-theory of a real semisimple Lie group which acts on the (complex) flag variety of its complexification group. We construct an assemble map in the framework of KK-theory and then we prove that it is an isomorphism. The prove relies on a careful study of the orbits of the real group action on the flag variety and then piecing together different orbits. This result is a special case of the Baum-Connes conjecture with coefficients.
Suppose $L$ is a semisimple Levi subgroup of a connected Lie group~$G$, $X$ is a Borel $G$-space with finite invariant measure, and $α\colon X \times G \to \GL_n(\real)$ is a Borel cocycle. Assume $L$ has finite center, and that the real rank of every simple factor of~$L$ is at least two. We show that if $L$ is ergodic on~$X$, and the restriction of~$α$ to~$X \times L$ is cohomologous to a homomorphism (modulo a compact group), then, after passing to a finite cover of~$X$, the cocycle $α$ itself is cohomologous to a homomorphism (modulo a compact group).
We prove that the Harish-Chandra--Schwartz space associated with a discrete subgroup of a semisimple Lie group is a dense subalgebra of the reduced $C^*$-algebra of the discrete subgroup. Then, we prove that for the reduced $C^*$-norm is controlled by the norm of the Harish-Chandra--Schwartz space. This inequality is weaker than property RD and holds for any discrete group acting isometrically, properly on a Riemannian symmetric space.
On the Schwartz estimate for Hodge Laplacians on semisimple Lie groups
本文将对称空间上的Schwartz估计推广到半单李群，证明了Hodge Laplacian与Dirac算子的相应估计，并给出李代数上同调的Kuga引理版本，以利用Fourier分析研究向量丛截面的热问题。
We determine a substantial part of the unitary representation theory of the Drinfeld double of a $q$-deformation of a compact Lie group in terms of the complexification of the compact Lie group. Using this, we show that the dual of every $q$-deformation of a higher rank compact Lie group has central property ${\rm (T)}$. We also determine the unitary dual of $SL_q(n,\mathbb{C})$.
We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical Symplectic reflection algebras of Etingof and Ginzburg. We further apply our classification and a classification of filtered Poisson deformations obtained by Namikawa to establish a version of the Orbit method for semisimple Lie algebras. Namely, we produce a natural map from the set of adjoint orbits in a semisimple Lie algebra to the set of primitive ideals in the universal enveloping algebra. We show that the map is in…
From the Levi's Theorem it is known that every finite dimensional Lie algebra over a field of characteristic zero is decomposed into semidirect sum of solvable radical and semisimple subalgebra. Moreover, semisimple part is the direct sum of simple ideals. In \cite{Bar} the Levi's theorem is extended to the case of Leibniz algebras. In the present paper we investigate the semisimple Leibniz algebras and we show that the splitting theorem for semisimple Leibniz algebras is not true. Moreover, we consider some special classes of the semisimple Leibniz algebras and find a condition under which th…
A semisimple algebraic tensor category over an algebraically closed field k of characteristic zero is the representation category of all finite dimensional twisted super representations of an affine reductive supergroup G over k. Such a supergroup is reductive if and only if its connected component is reductive. The connected component is reductive if and only if the Lie superalgebra divided by its center is a product of simple Lie algebras of classical type and Lie superalgebras spo(1,2r) of the orthosymplectic types BC_r.
The present paper is devoted to the description of finite-dimensional semisimple Leibniz algebras over complex numbers, their derivations and automorphisms.
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。