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 $\mathfrak{g}$.
In this paper we study the isotypic decomposition of the regular module of a finite-dimensional Hopf algebra over an algebraically closed field of characteristic zero. For a semisimple Hopf algebra, the idempotents realizing the isotypic decomposition can be explicitly expressed in terms of characters and the Haar integral. In this paper we investigate Hopf algebras with the Chevalley property, which are not necessarily semisimple. We find explicit expressions for idempotents in terms of Hopf-algebraic data, where the Haar integral is replaced by the regular character of the dual Hopf algebra.…
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 $A=S+N$.
In this paper, we consider the Drinfeld double $\D$ of a $12$-dimensional Hopf algebra $\C$ over an algebraically closed field of characteristic zero whose coradical is not a subalgebra and describe its simple modules, projective covers of the simple modules and show that it is of wild representation type. Moreover, we show that the Nichols algebras associated to non-simple indecomposable modules are infinite-dimensional. In particular, for any object $V$ in $\CYD$, if $\BN(V)$ is finite-dimensional, then $V$ must be semisimple. Finally, we describe the Nichols algebras associated to partial s…
We consider the Chevalley involution in the context of real reductive groups. We show that if G(R) is the real points of a connected reductive group, there is an involution, unique up to conjugacy by G(R), taking any semisimple element to a conjugate of its inverse. As applications we give a condition for every irreducible representation of G(R) to be self-dual, and to the Frobenius Schur indicator for such groups.
Let $\g$ be a finite dimensional Lie algebra over a field $k$ of characteristic zero. An element $x$ of $\g$ is said to have an \emph{abstract Jordan-Chevalley decomposition} if there exist unique $s,n\in\g$ such that $x=s+n$, $[s,n]=0$ and given any finite dimensional representation $π:\g\to\gl(V)$ the Jordan-Chevalley decomposition of $π(x)$ in $\gl(V)$ is $π(x)=π(s)+π(n)$. In this paper we prove that $x\in\g$ has an abstract Jordan-Chevalley decomposition if and only if $x\in [\g,\g]$, in which case its semisimple and nilpotent parts are also in $[\g,\g]$ and are explicitly determined. We d…
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.
In this paper, we describe a connection that exists among (a) the number of singular points along the trajectory of Toda flow, (b) the cohomology of a compact subgroup $K$, and (c) the number of points of a Chevalley group $K({\mathbb F}_q)$ related to $K$ over a finite field ${\mathbb F}_q$. The Toda lattice is defined for a real split semisimple Lie algebra $\mathfrak g$, and $K$ is a maximal compact Lie subgroup of $G$ associated to $\mathfrak g$. Relations are also obtained between the singularities of the Toda flow and the integral cohomology of the real flag manifold $G/B$ with $B$ the B…
Let (g,[p]) be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p>0. The scheme of tori of maximal dimension of g gives rise to a finite group S(g) that coincides with the Weyl group of g in case g is a Lie algebra of classical type. In this paper, we compute the group S(g) for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants.
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}…
Let $g$ be a complex semisimple Lie algebra with adjoint group $G$. Suppose that $σ$ is an involutive automorphism of $g$. Then $σ$ induces uniquely an involution of $G$ also denoted by $σ$, let $K=G^σ$ be a subgroup of $σ$-fixed points. Consider a direct decomposition $g=k+p$ of $g$ into eigenspaces for $σ$. Then $p$ is a $K$-module. Denote by $a\subset p$ any maximal abelian ad-diagonalizable subalgebra. Consider the ``baby Weyl group'' $W=N_K(a)/Z_K(a)$. Let $ψ: C[p]^K\to C[a]^W$ be a restriction map of algebras of invariants. Then the famous Chevalley restriction theorem states that $ψ$ is…
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
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.
Let $\mathsf G$ be a connected reductive linear algebraic group defined over $\mathbb R$, and let $C: \mathsf G\rightarrow \mathsf G$ be a fundamental Chevalley involution. We show that for every $g\in \mathsf G(\mathbb R)$, $C(g)$ is conjugate to $g^{-1}$ in the group $\mathsf G(\mathbb R)$. Similar result on the Lie algebras is also obtained.
The traditional construction of Chevalley groups relies on the choice of certain signs for a Chevalley basis of the underlying Lie algebra~$\mathfrak{g}$. Recently, Lusztig simplified this construction for groups of adjoint type by using the "canonical basis" of the adjoint representation of~$\mathfrak{g}$, in particular, no choices of signs are required. The purpose of this note is to extend this to Chevalley groups which are not necessarily of adjoint type, using Jantzen's explicit models of the minuscule highest weight representations of~$\mathfrak{g}$.
Let $G$ be a simply-connected semisimple algebraic group scheme over an algebraically closed field of characteristic $p > 0$. Let $r \geq 1$ and set $q = p^r$. We show that if a rational $G$-module $M$ is projective over the $r$-th Frobenius kernel $G_r$ of $G$, then it is also projective when considered as a module for the finite subgroup $\Gfq$ of $\Fq$-rational points in $G$. This salvages a theorem of Lin and Nakano (\emph{Bull.\ London Math.\ Soc.} 39 (2007) 1019--1028). We also show that the corresponding statement need not hold when the group $G$ is replaced by the unipotent radical $U$…
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.
Chevalley's theorem states that for any simple finite dimensional Lie algebra G (1) the restriction homomorphism of the algebra of polynomials on G onto the Cartan subalgebra H induces an isomorphism between the algebra of G-invariant polynomials on G with the algebra of W-invariant polynomals on H, where W is the Weyl group of G, (2) each G-invariant polynomial is a linear combination of the powers of traces tr r(x), where r is a finite dimensional representation of G. None of these facts is necessarily true for simple Lie superalgebras. We reformulate Chevalley's theorem so as to embrace Lie…
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.
In the framework of algebraic supergeometry, we give a construction of the scheme-theoretic supergeometric analogue of Chevalley groups, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. In particular both basic (even exceptional) and strange cases are covered. This provides a unified approach to most of the algebraic supergroups considered so far in literature, and an effective method to construct new ones. As an intermediate step, we prove an existence theorem for Chevalley bases of simple classical Lie superalgebras and a PBW-like theorem for thei…
An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical. Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic, and some other cases. Let $G$ ba a connected semi-reductive algebraic group over an algebraically closed field $\mathbb{F}$ and $\mathfrak{g}=Lie(G)$. It turns out that $G$ has many same properties as reductive groups, such as the Bruhat decomposition. In this note, we…
We prove the Chevalley restriction theorem for the commuting scheme of symplectic Lie algebras. The key step is the construction of the inverse map of the Chevalley restriction map called the spectral data map. Along the way, we establish a certain multiplicative property of the Pfaffian which is of independent interest.
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.
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…
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.
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 $G.$
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 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.
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.
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.
An adjoint Chevalley group of rank at least 2 over a rational algebra (or a similar ring), its elementary subgroup, and the corresponding Lie ring have the same automorphism group. These automorphisms are explicitly described.
Let p be a maximal truncated parabolic subalgebra of a simple Lie Algebra. It was shown in many cases that the Poisson centre Y(p) is a polynomial algebra. We construct a slice for the coadjoint action of p, thus extending a theorem of Kostant. The role of the principal sl_2-triple is played by an adapted pair.
For 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 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…
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.
Let $G$ be the symplectic group, $Φ=C_n$ its root system, $B\subset G$ its standard Borel subgroup, $W$ the Weyl group of $Φ$. To each involution $σ\in W$ one can assign the $B$-orbit $Ω_σ$ contained in the dual space of the Lie algebra of the unipotent radical of $B$. We prove that $Ω_σ$ is contained in the Zariski closure of $Ω_τ$ if and only of $σ\leqτ$ with respect to the Bruhat--Chevalley order. We also prove that $\dimΩ_σ$ is equal to $l(σ)$, the length of $σ$ in $W$.
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.
