DOCSLIB.ORG

Of Parabolic and Seaweed Subalgebras Of

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Of Parabolic and Seaweed Subalgebras Of View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Linear Algebra and its Applications 374 (2003) 127–142 www.elsevier.com/locate/laa Index of parabolic and seaweed subalgebras ୋ of son Alexander Dvorsky Department of Mathematics, University of Miami, P.O. Box 249085, Coral Gables, FL 33124, USA Received 3 December 2002; accepted 23 April 2003 Submitted by H. Schneider Abstract The index of a Lie algebra r is defined as minw∈r∗ dim rw, where rw is the stabilizer of w ∗ under the coadjoint action of r on r . In this paper we derive simple formulas for the index of parabolic subalgebras of the even orthogonal algebra g = so(2n, k). We use these formulas to prove the conjecture that ind q <nfor any parabolic subalgebra or seaweed subalgebra (inter- section of two weakly opposite parabolics) q of so(2n, k). Some partial results for subalgebras of g = so(2n + 1, k) are also obtained. © 2003 Elsevier Inc. All rights reserved. AMS classification: 17B20; 17B10 Keywords: Parabolic subalgebras; Seaweed subalgebras; Lie algebra index; Frobenius Lie algebra 1. Introduction Let r be a Lie algebra over the algebraically closed field k,andV beafinite- dimensional r-module. We call an element v ∈ V regular if the orbit r · v has the maximal possible dimension (equivalently, the stabilizer rv has the minimal dimen- sion). Consider the dual module V ∗. Denote ind(r,V):= dim V − max (dim r · w). (1) w∈V ∗ ୋ Research supported in part by NSF grant no. 0202132. E-mail address: [email protected] (A. Dvorsky). 0024-3795/$ - see front matter 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0024-3795(03)00552-4 128 A. Dvorsky / Linear Algebra and its Applications 374 (2003) 127–142 This is the index of the r-module V . In the special case of the adjoint representa- tion (V = r), we simply write ind r, and speak about the index of a Lie algebra r.In other words, ind r = min dim rw, (2) w∈r∗ ∗ where rw is the stabilizer of w under the coadjoint action of r on r . The notion of an index of a Lie algebra was introduced by Diximier in [2]. Com- puting the index is trivial for reductive Lie algebras, because in this case the index simply equals the rank of the algebra. On the other hand, for non-reductive Lie alge- bras the index is often hard to estimate––for example, the index for a nilpotent r is equal to the Gelfand–Kirillov dimension of the center of the enveloping algebra of r. The explicit results are known only for some special cases [4,7]. Recently there was a renewal of the interest in the problems of finding indices for various subalgebras in semisimple Lie algebras, motivated by the questions in representation theory and invariant theory. Parabolic subalgebras and their generalizations form one of the most interesting classes of non-reductive Lie algebras. The seaweed subalgebras were introduced in [1] as the intersections of two parabolic subalgebras whose sum is all of g (say, for g = gl(n, k) the seaweed subalgebras are those subalgebras that preserve a pair of opposite partial flags in the vector space kn). In [6], Panyushev proposed the following conjecture: Conjecture 1.1. Let g be a reductive Lie algebra over the algebraically closed field k.Then: (a) For any nonreductive seaweed subalgebra s in g,inds < rank g. (b) For any proper parabolic subalgebra p in q,indp < rank g. The second part of the conjecture is obviously just the corollary of the first part, since any parabolic subalgebra is also a seaweed subalgebra. On the other hand, it was shown in [6] that for classical Lie algebras the second part of the conjecture actually implies the first. Clearly, to prove the above Conjecture it suffices to verify it for the subalge- bras in simple g. In [1], the index of a seaweed subalgebra in gl(n) was expressed in terms of invariants of a certain graph (“meander graph”). This is sufficient to settle the conjecture for g of type An. The reductive procedure introduced in [6] was powerful enough to confirm the Conjecture for all parabolic subalgebras in Lie algebras of type Cn (i.e., g = sp(2n, k))andsome parabolic subalgebras in algebras of type Bn, Dn (g = so(m, k)). The conjecture for the general parabolic subalgebras in orthogonal algebras (as well as for the exceptional simple g) remained open. The main difficulties in extending the results of [1,6] to the orthogonal case are essentially computational. We shall see that the problem of finding the index of an A. Dvorsky / Linear Algebra and its Applications 374 (2003) 127–142 129 arbitrary parabolic p in so(2n) can be reduced to a linear algebra problem. Namely, we will need to look for normal forms of matrices under suitable conjugations and describe the stabilizers for these normal forms. In this paper we will combine the approach of [6,7] with some computational results of [3] to: (a) give explicit formulas for ind p,wherep is a parabolic subalgebra of so(2n, k), (b) prove Conjecture 1.1 for g = so(2n, k). We will also derive expressions for ind p, for many (but not all) parabolic and seaweed subalgebras p ⊂ so(2n + 1, k). 2. Preliminaries We use the following conventions throughout the paper: • All Lie algebras are defined over k,wherek is algebraically closed field of char- acteristic 0. • Capital letters are used to denote Lie groups and the corresponding fractur letters are used for their Lie algebras. • The omitted entries in matrices are always assumed to be 0. We write zt for a trans- pose of z, and use the notation diag(z1,z2,...) for the block-diagonal matrices with blocks z1,z2,... • We write a, b etc. for the ordered strings of positive integers. 2.1. General properties of ind r We list the basic properties of ind(r,V)and ind r for future reference. All of the material in this section is well known. These standard facts are obvious from the definition of the index: • ind(r,V1 ⊕V2) = ind(r,V1)+ind(r,V2), in particular ind r1 ⊕r2 = ind r1 +ind r2, • ind r = rank r for r reductive, • ind r = dim r for r abelian. The following proposition follows immediately from [7] (see also [6, 1.4]). Proposition 2.1. Let r = q + a be a semidirect sum of a Lie algebra q and an ab- elian ideal a.LetQ be a connected Lie group with Lie algebra q, and assume that Q has an open coadjoint orbit Q · w in a∗.Then = ind r ind qw, where qw is the Lie algebra of a stabilizer Qw. 130 A. Dvorsky / Linear Algebra and its Applications 374 (2003) 127–142 Proof. Directly from the definition of index, in this case ind(q, a) = 0. Observe that the open orbit Q · w contains all q-regular elements in a∗. The theorem of Raïs [7] on the index of semidirect sums then gives + = + = ind(q a) ind(q, a) ind qw ind qw. Let now r = r(0) + r(1) + r(2) be a grading of r (i.e., a Z-grading of height 2). We shall refer to a grading like this as a 3-grading. A 3-grading of r induces the grading r∗ = r∗(−2) + r∗(−1) + r∗(0) on r∗, with r∗(−2) = Ann(r(0) + r(1)) etc. Proposition 2.2 [6, 1.5]. Let w be a r(0)-regular element in r∗(−2), and suppose that r · w/= w for r ∈ r(1), r/= 0.Then ind r = ind r(0)w + ind(r(0), r(2)). (3) Remark. Assume that r = r(1) + r(2) is a nilpotent Lie algebra with center r(2) (in the present paper, this will always be the case). Let R be a connected nilpotent Lie group with Lie algebra r. Call a representation of R generic if it corresponds (in the sense of Kirillov’s orbit method) to some regular element of (r)∗. Then the conditions of the Proposition above are satisfied if the generic representations of the group R are determined by their restrictions on the center Z(R) = exp r(2). 2.2. so(2n) and its parabolic subalgebras By SO(2n) = SO(2n, k) we understand the special orthogonal group, realized as the group of all x ∈ SL(2n, k) satisfying 0 I 0 I xt n x = n . In 0 In 0 Then the Lie algebra g = so(2n) consists of the matrices z11 z12 − t , z21 z11 2 n with z12,z21 skew-symmetric n × n matrices (we write z12,z21 ∈ Skew(n) = k ), n×n and z11 ∈ k is arbitrary. We realize the symplectic group Sp(2n) as the group of all x ∈ GL(2n, k) satis- fying xt J x = J ,where 2n 2n 0 In J2n = −In 0 and write sp(2n) for its Lie algebra. Fix the Cartan subalgebra t in g consisting of the diagonal matrices: t = {diag(a1,a2,...,an, −a1, −a2,...,−an)} A. Dvorsky / Linear Algebra and its Applications 374 (2003) 127–142 131 and consider the triangular decomposition g = u + t + u , with u corresponding to taking z21 = 0andz11 upper triangular. Declare the roots of t in u positive, and let ={α1,α2,...,αn} be the corresponding set of simple roots. Here αi = εi − εi+1 (1 i n − 1) and αn = εn−1 + εn,whereεi(diag(a1,a2,..., an, −a1, −a2,...,−an)) = ai.
Load more

Recommended publications
  • The Unbroken Spectrum of Frobenius Seaweeds II: Type-B and Type-C
    The unbroken spectrum of Frobenius seaweeds II: type-B and type-C Alex Cameron∗, Vincent E. Coll, Jr.∗∗, Matthew Hyatt†, and Colton Magnant†† July 23, 2019 ∗Department of Mathematics, Lehigh University, Bethlehem, PA, USA: [email protected] ∗∗Department of Mathematics, Lehigh University, Bethlehem, PA, USA: [email protected] †FactSet Research Systems, New York, NY, USA: [email protected] †† Department of Mathematics, Clayton State University, Morrow, GA, USA: [email protected] Abstract Analogous to the Type-An−1 = sl(n) case, we show that if g is a Frobenius seaweed subalgebra of Bn = so(2n +1) or Cn = sp(2n), then the spectrum of the adjoint of a principal element consists of an unbroken set of integers whose multiplicities have a symmetric distribution. Mathematics Subject Classification 2010 : 17B20, 05E15 Key Words and Phrases: Frobenius Lie algebra, seaweed, biparabolic, principal element, Dynkin diagram, spectrum, regular functional, Weyl group 1 Introduction Notation: All Lie algebras will be finite dimensional over C, and the Lie multiplication will be denoted by [-,-]. The index of a Lie algebra is an important algebraic invariant and, for seaweed algebras, is bounded by the algebra’s rank: ind g ≤ rk g, (see [11]). More formally, the index of a Lie algebra g is given by arXiv:1907.08775v1 [math.RT] 20 Jul 2019 ind g = min dim(ker(BF )), F ∈g∗ where F is a linear form on g, and BF is the associated skew-symmetric bilinear Kirillov form, defined by BF (x,y) = F ([x,y]) for x,y ∈ g. On a given g, index-realizing functionals are called regular and exist in profusion, being dense in both the Zariski and Euclidean topologies of g∗.
    [Show full text]
  • (Bi)Parabolic Subalgebras in the Reductive Lie Algebras. Karin Baur, Anne Moreau
    Quasi-reductive (bi)parabolic subalgebras in the reductive Lie algebras. Karin Baur, Anne Moreau To cite this version: Karin Baur, Anne Moreau. Quasi-reductive (bi)parabolic subalgebras in the reductive Lie algebras.. 2008. hal-00348974v1 HAL Id: hal-00348974 https://hal.archives-ouvertes.fr/hal-00348974v1 Preprint submitted on 22 Dec 2008 (v1), last revised 30 Jun 2010 (v2) HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. QUASI-REDUCTIVE (BI)PARABOLIC SUBALGEBRAS IN THE REDUCTIVE LIE ALGEBRAS. KARIN BAUR AND ANNE MOREAU Abstract. Let g be a finite dimensional Lie algebra, and z its center. We say that g is quasi- reductive if there is f ∈ g∗ such that g(f)/z is a reductive Lie algebra whose center consists of semisimple elements, where g(f) denotes the stabilizer of f in g for the coadjoint action. If g is reductive, then g is quasi-reductive itself, and also every Borel or Levi subalgebra of g. However the parabolic subalgebras of g are not always quasi-reductive (except in types A or C, see [P03]). Biparabolic (or seaweed) subalgebras are the intersection of two parabolic subalgebras whose sum is g.
    [Show full text]
  • The Derived Algebra of a Stabilizer, Families of Coadjoint Orbits, and Sheets, Arxiv:1202.1135V2 (2012)
    The derived algebra of a stabilizer, families of coadjoint orbits, and sheets Anton Izosimov∗ Abstract Let g be a finite-dimensional real or complex Lie algebra, and let µ ∈ g∗. In the first part of the paper, the relation is discussed between the derived algebra of the stabilizer of µ and the set of coadjoint orbits which have the same dimension as the orbit of µ. In the second part, semisimple Lie algebras are considered, and the relation is discussed between the derived algebra of a centralizer and sheets. 1 Introduction Let g be a finite-dimensional real or complex Lie algebra. The group G acts on the dual space g∗ via the coadjoint action, and g∗ is foliated into the orbits of this action. Consider the union of all orbits ∗ ∗ which have the same codimension k. Denote this union by gk. Each of the sets gk is a quasi-affine ∗ algebraic variety. The study of the varieties gk was initiated by A.Kirillov in connection with the orbit method [6], which relates the unitary dual of G to the set of coadjoint orbits g∗/G. The sets ∗ ∗ gk/G appear in this picture as natural strata of g /G, therefore it is important to understand the ∗ geometry of gk for each k. ∗ So, consider a finite-dimensional real or complex Lie algebra g. Let gµ = {x ∈ g | ad x(µ)=0} be the stabilizer of an element µ ∈ g∗ with respect to the coadjoint representation of g. In the present note, the following simple geometric fact is proved: any element ξ ∈ g∗ which is tangent ∗ ∗ to the variety gk at a point µ ∈ g vanishes on the derived algebra of gµ.
    [Show full text]
  • Periodic Automorphisms of Takiff Algebras, Contractions, and Θ-Groups
    October 15, 2007 PERIODIC AUTOMORPHISMS OF TAKIFF ALGEBRAS, CONTRACTIONS, AND θ-GROUPS DMITRI I. PANYUSHEV INTRODUCTION Let G be a connected reductive algebraic group with Lie algebra g. The ground field | is algebraically closed and of characteristic zero. Fundamental results in invariant theory of the adjoint representation of G are primarily associated with C. Chevalley and B. Kostant. Especially, one should distinguish the ”Chevalley restriction theorem” and seminal article of Kostant [5]. Later, Kostant and Rallis extended these results to the isotropy representa- tion of a symmetric variety [6]. In 1975, E.B. Vinberg came up with the theory of θ-groups. This theory generalises and presents in the most natural form invariant-theoretic results previously known for the adjoint representation and isotropy representations of the sym- metric varieties. Let us remind the main construction and results of Vinberg's article [15]. Let θ Aut(g) 2 be a periodic (= finite order) automorphism of g. The order of θ is denoted by θ . Fix a jθj j j primitive root of unity ζ = p1 and consider the periodic grading (or Zjθj-grading) g = gi ; i Z M2 jθj i θ where gi is the ζ -eigenspace of θ. In particular, g0 = g is the fixed point subalgebra for θ. Let G0 be the connected subgroup of G with Lie algebra g0. The restriction of the adjoint representation yields the natural homomorphism G GL(g ). The linear groups 0 ! 1 obtained in this way are called θ-groups, and the point is that they have the best possible invariant-theoretic properties: |[g ]G0 is a polynomial algebra; • 1 the quotient morphism π : g g ==G = Spec(|[g ]G0 ) is flat; • 1 ! 1 0 1 each fibre of π contains finitely many G -orbits.
    [Show full text]
  • Coadjoint Orbits of Lie Algebras and Cartan Class
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 002, 20 pages Coadjoint Orbits of Lie Algebras and Cartan Class Michel GOZE y and Elisabeth REMM z y Ramm Algebra Center, 4 rue de Cluny, F-68800 Rammersmatt, France E-mail: [email protected] z Universit´ede Haute-Alsace, IRIMAS EA 7499, D´epartement de Math´ematiques, F-68100 Mulhouse, France E-mail: [email protected] Received September 13, 2018, in final form December 31, 2018; Published online January 09, 2019 https://doi.org/10.3842/SIGMA.2019.002 Abstract. We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit O(α) at the point α corresponds to the characteristic space associated to the left invariant form α and its dimension is the even part of the Cartan class of α. We apply this remark to determine Lie algebras such that all the nontrivial orbits (nonreduced to a point) have the same dimension, in particular when this dimension is 2 or 4. We determine also the Lie algebras of dimension 2n or 2n + 1 having an orbit of dimension 2n. Key words: Lie algebras; coadjoint representation; contact forms; Frobenius Lie algebras; Cartan class 2010 Mathematics Subject Classification: 17B20; 17B30; 53D10; 53D05 1 Introduction Let G be a connected Lie group, g its Lie algebra and g∗ the dual vector space of g. We identify g with the Lie algebra of left invariant vector fields on G and g∗ with the vector space of left invariant Pfaffian forms on G.
    [Show full text]
  • Invariant Nijenhuis Tensors and Integrable Geodesic Flows
    Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 056, 30 pages Invariant Nijenhuis Tensors and Integrable Geodesic Flows Konrad LOMPERT y and Andriy PANASYUK z y Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662 Warszawa, Poland E-mail: [email protected] z Faculty of Mathematics and Computer Science, University of Warmia and Mazury, ul. Sloneczna 54, 10-710 Olsztyn, Poland E-mail: [email protected] Received December 19, 2018, in final form August 02, 2019; Published online August 07, 2019 https://doi.org/10.3842/SIGMA.2019.056 Abstract. We study invariant Nijenhuis (1; 1)-tensors on a homogeneous space G=K of a reductive Lie group G from the point of view of integrability of a Hamiltonian system of differential equations with the G-invariant Hamiltonian function on the cotangent bun- ∗ ∗ dle T (G=K). Such a tensor induces an invariant Poisson tensor Π1 on T (G=K), which is Poisson compatible with the canonical Poisson tensor ΠT ∗(G=K). This Poisson pair can be reduced to the space of G-invariant functions on T ∗(G=K) and produces a family of Poisson commuting G-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integra- bility in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces G=K of compact Lie groups G for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of G to two subgroups G = G1 · G2, where G=Gi are symmetric spaces, K = G1 \ G2.
    [Show full text]
  • Coadjoint Orbits of Reductive Type of Seaweed Lie Algebras
    COADJOINT ORBITS OF REDUCTIVE TYPE OF SEAWEED LIE ALGEBRAS ANNE MOREAU AND OKSANA YAKIMOVA ABSTRACT. A connected algebraic group Q defined over a field of characteristic zero is quasi-reductive if there is an element of q∗ of reductive type, that is such that the quotient of its stabiliser by the centre of Q is a reductive subgroup of GL(q). Such groups appear in harmonic analysis when unitary representations are studied. In particular, over the field of real numbers they turn out to be the groups with discrete series and their irreducible unitary square integrable representations are parameterised by coadjoint orbits of reductive type. Due to results of M. Duflo, coadjoint representation of a quasi- reductive Q possesses a so called maximal reductive stabiliser and knowing this subgroup, defined up to a conjugation in Q, one can describe all coadjoint orbits of reductive type. In this paper, we consider quasi-reductive parabolic subalgebras of simple complex Lie algebras as well as seaweed subalgebras of gln(C) and describe the classes of their maximal reductive stabilisers. 1. INTRODUCTION 1.1. Suppose that k is a field of characteristic zero and Q a connected algebraic (or Lie) group defined over k. Let q = Lie Q be the Lie algebra of Q. Let Z denote the centre of Q. A linear function γ ∈ q∗ is said to be of reductive type if the quotient Qγ /Z of the stabiliser Qγ ⊂ Q for the coadjoint action is a reductive subgroup of GL(q∗) (or, what is equivalent, of GL(q)). Whenever it makes sense, we will also say that γ is of compact type if Qγ /Z is compact.
    [Show full text]
  • Mathematisches Forschungsinstitut Oberwolfach
    Mathematisches Forschungsinstitut Oberwolfach Report No. 21/2012 DOI: 10.4171/OWR/2012/21 Toric Geometry Organised by Klaus Altmann, Berlin Victor Batyrev, T¨ubingen Yael Karshon, Toronto April 15th – April 21st, 2012 Abstract. Toric Geometry plays a major role where a wide variety of math- ematical fields intersect, such as algebraic and symplectic geometry, algebraic groups, and combinatorics. The main feature of this workshop was to bring people from these area together to learn about mutual, possibly up till now unnoticed similarities in their respective research. Mathematics Subject Classification (2000): 14M25, 53D37, 52B20. Introduction by the Organisers The workshop “Toric Geometry” was attended by 53 people including many young participants. The idea was to shed light on the subject from many different points of view – toric geometry involves methods from algebraic and symplectic geometry, algebraic groups, and discrete mathematics. A major driving force combining all these directions is still provided by the dif- ferent flavours of mirror symmetry. So it is quite natural that related subjects like Lagrangians in symplectic manifolds showed up in many talks (Abreu, Woodward, Ono, Lau, Sjamaar). A very common feature that appeared in many talks was the attempt to weaken assumptions in the setting of algebraic or symplectic toric varieties. This was done by either considering higher complexities of torus actions, or by relaxing the demands on the symplectic forms, or by studying non-algebraic situations or more general algebraic groups than just tori (Timashev, Hausen, S¨uß, Knop, Tolman, Holm, Masuda). Polyhedral methods and their interplay with resolutions and deformations or degenerations is a classical feature of toric geometry.
    [Show full text]
  • Reduction by Stages and the Ra¨Is-Type Formula for the Index of Lie Algebra Extensions
    Noname manuscript No. (will be inserted by the editor) Andriy Panasyuk Reduction by stages and the Ra¨ıs-type formula for the index of Lie algebra extensions 0 Introduction The index of a Lie algebra is defined as the dimension of the stabilizer of a generic element with respect to the coadjoint representation, or equivalently, the codimension of a generic coadjoint orbit. The index is an important characteristic of a Lie algebra, which is used in different applications. The aim of this note is to give a generalization of the so-called Ra¨ıs formula for the index of a semidirect product of a Lie algebra and an abelian ideal. It is well known that the index of a semisimple Lie algebra coincides with its rank. For the nonsemisimple case, one of the most popular in applications related result is the above mentioned Ra¨ıs formula calculating the index of a semidirect product s = g ×ρ V of a Lie algebra g and a vector space V , by means of the representation ρ∗ : g → gl(V ∗) dual to the representation ρ ([Rai78], see also Corollary 1.8). More precisely, the index of s is equal to the sum of the codimension of a generic orbit of ρ∗ and the index of the stabilizer with respect to ρ∗ of a generic element in V ∗. The Ra¨ıs formula, first proved by purely algebraic methods, can be also deduced from Poisson geometric results on semidirect products. Namely, the following theorem could be found in [RSTS94, Section 5]. If s = g ×ρ V is a ∗ ∗ semidirect product and Vν := g × Oν , where Oν ⊂ V is the G-orbit of any ∗ ∗ element ν ∈ V , then: 1) Vν is a Poisson submanifold in (g ×ρ V ) ; 2) Vν is ∗ Poisson diffeomorphic to T G/Gν (here G ⊃ Gν are the Lie groups corre- ∗ sponding to the Lie algebras g, gν, where gν is the stabilizer of ν, T G/Gν is endowed with the Poisson structure being the reduction of the canonical one).
    [Show full text]
  • Coadjoint Orbits of Lie Algebras and Cartan Class Michel Goze, Elisabeth Remm
    Coadjoint Orbits of Lie Algebras and Cartan Class Michel Goze, Elisabeth Remm To cite this version: Michel Goze, Elisabeth Remm. Coadjoint Orbits of Lie Algebras and Cartan Class. Symmetry, Integrability and Geometry : Methods and Applications, National Academy of Science of Ukraine, 2019, 15, pp.002. 10.3842/SIGMA.2019.002. hal-02383703 HAL Id: hal-02383703 https://hal.archives-ouvertes.fr/hal-02383703 Submitted on 27 Nov 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 002, 20 pages Coadjoint Orbits of Lie Algebras and Cartan Class Michel GOZE y and Elisabeth REMM z y Ramm Algebra Center, 4 rue de Cluny, F-68800 Rammersmatt, France E-mail: [email protected] z Universit´ede Haute-Alsace, IRIMAS EA 7499, D´epartement de Math´ematiques, F-68100 Mulhouse, France E-mail: [email protected] Received September 13, 2018, in final form December 31, 2018; Published online January 09, 2019 https://doi.org/10.3842/SIGMA.2019.002 Abstract. We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit O(α) at the point α corresponds to the characteristic space associated to the left invariant form α and its dimension is the even part of the Cartan class of α.
    [Show full text]
  • Arxiv:Math/0506579V2
    October 17, 2007 SEMI-DIRECT PRODUCTS OF LIE ALGEBRAS AND THEIR INVARIANTS DMITRI I. PANYUSHEV Independent University of Moscow, Bol’shoi Vlasevskii per. 11 119002 Moscow, Russia e-mail: [email protected] CONTENTS Introduction 1 1. Preliminaries 6 2. Generic stabilisers (centralisers) for the adjoint representation 8 3. Generic stabilisers for the coadjoint representation 10 4. Semi-direct products of Lie algebras and modules of covariants 12 5. Generic stabilisers and rational invariants for semi-direct products 14 6. Reductive semi-direct products and their polynomial invariants 20 7. Takiff Lie algebras and their invariants 23 8. The null-cone and its irreducibility 25 9. Isotropy contractions and Z2-contractions of semisimple Lie algebras 30 10. Reductive Takiff Lie algebras and their representations 38 11. On invariants and null-cones for generalised Takiff Lie algebras 44 References 47 INTRODUCTION arXiv:math/0506579v2 [math.AG] 17 Oct 2007 The ground field k is algebraically closed and of characteristic zero. The goal of this pa- per is to extend the standard invariant-theoretic design, well-developed in the reductive case, to the setting of non-reductive group representations. This concerns the follow- ing notions and results: the existence of generic stabilisers and generic isotropy groups for (finite-dimensional rational) representations; structure of the fields and algebras of invariants; quotient morphisms and structure of their fibres. One of the main tools for obtaining non-reductive Lie algebras is the semi-direct product construction. There is a number of articles devoted to the study of the coadjoint representations of non-reductive 2000 Mathematics Subject Classification.
    [Show full text]
  • Sugawara Operators for Classical Lie Algebras
    Mathematical Surveys and Monographs Volume 229 Sugawara Operators for Classical Lie Algebras Alexander Molev 10.1090/surv/229 Sugawara Operators for Classical Lie Algebras Mathematical Surveys and Monographs Volume 229 Sugawara Operators for Classical Lie Algebras Alexander Molev EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 17B35, 17B63, 17B67, 17B69, 16S30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-229 Library of Congress Cataloging-in-Publication Data Names: Molev, Alexander, 1961- author. Title: Sugawara operators for classical Lie algebras / Alexander Molev. Description: Providence, Rhode Island : American Mathematical Society, [2018] | Series: Mathe- matical surveys and monographs ; volume 229 | Includes bibliographical references and index. Identifiers: LCCN 2017041529 | ISBN 9781470436599 (alk. paper) Subjects: LCSH: Lie algebras. | Affine algebraic groups. | Kac-Moody algebras. | AMS: Nonas- sociative rings and algebras – Lie algebras and Lie superalgebras – Universal enveloping (su- per)algebras. msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Poisson algebras. msc | Nonassociative rings and algebras – Lie algebras and Lie superalgebras – Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras. msc | Nonas- sociative rings and algebras – Lie algebras and Lie superalgebras – Vertex operators; vertex operator algebras and related structures. msc | Associative rings and algebras – Rings and algebras arising under various constructions – Universal enveloping algebras of Lie algebras. msc Classification: LCC QA252.3 .M6495 2018 | DDC 512/.482–dc23 LC record available at https://lccn.loc.gov/2017041529 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research.
    [Show full text]