A Survey on Invariant Cones in Infinite Dimensional Lie Algebras
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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. -
C*-Algebras and Kirillov's Coadjoint Orbit Method
C∗-ALGEBRAS AND KIRILLOV'S COADJOINT ORBIT METHOD DAVID SCHWEIN One of the main goals of representation theory is to understand the unitary dual of a topological group, that is, the set of irreducible unitary representations. Much of modern number theory, for instance, is concerned with describing the unitary duals of various reduc- tive groups over a local field or the adeles, and here our understanding of the representation theory is far from complete. For a different class of groups, the nilpotent Lie groups, A. A. Kirillov gave in the mid- nineteenth century [Kir62] a simple and transparent description of the unitary dual: it is the orbit space under the coadjoint action of the Lie group on the dual of its Lie algebra. The goal of this article, notes for a talk, is to explain Kirillov's result and illustrate it with the Heisenberg group, following Kirillov's excellent and approachable book on the subject [Kir04]. We begin with an introductory section on the unitary dual of a C∗-algebra, the proper setting (currently) for unitary representations of locally compact groups, following Dixmier's exhaustive monograph on C∗-algebras [Dix77]. 1. C∗-algebras and the unitary dual The theory of unitary representations of locally compact topological groups, for instance, reductive p-adic groups, is a special case of the more general theory of representations of C∗-algebras. In this section we review the representation theory of C∗-algebras and see how it specializes to that of topological groups. 1.1. Definitions and examples. A Banach algebra is a Banach space A equipped with an algebra structure with respect to which the norm is sub-multiplicative: ka · bk ≤ kak · kbk; a; b 2 A: We do note require Banach algebras to be unital, and in fact, we will see shortly that there are many natural examples that are not unital. -
Representation Theory
M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23. -
The Solution to a Generalized Toda Lattice and Representation Theory
ADVANCES IN MATHEMATICS 34,195338 (1979) The Solution to a Generalized Toda Lattice and Representation Theory BERTRAM KOSTANT* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 CONTENTS 0. Introduction 0.1. Statement of the main results 0.2. A brief description of Sections l-7 0.3. General comments 1. Poisson commutatierity of translated invariants 1.1. A formula for the Poisson bracket 1.2. The case of a subalgebra of a semi-simple Lie algebra 1.3. Relating cent 8’ to S(g)’ 1.4. The theorem [I’, Jf3 = 0 for Lie summands 1.5. The case of a Bore1 subalgebra where f = C e-,. 1.6. The commutativity of certain vector fields on a;’ 2. The variety Z of normalized Jacobi elements 2.1. The variety 8, = f + d and invariants I E So 2.2. The vector fields II, I E So, on Z 2.3. The subvarieties Z(y) Z; Z and IV-conjugacy 2.4. The Bruhat-Gelfand decomposition, the isomorphism j3,: Q* -+ Z(r), and the rationality of Z(y) in the complex case 2.5. The foliation Z = U Z(r) in the complex case 2.6. The isomorphism b(W): G1”,, + Z(y) in the complex case 3. The parametrization Z s H x A, in the real case 3.1. The polar decomposition and centralizers # 3.2. The relation Gv n RHN = G,u for y E Z and the isomorphism Z(y) z [w’ 3.3. The open Weyl chamber R, and Z 3.4. The normalizer G of 9 in Ad gc and the subset G(*) = s(K)G* C G 3.5. -
Math.RT] 3 Feb 2006 .Tehiebr Group Heisenberg the 8
KYUNGPOOK Math J. 42(2002), 199-272 The Method of Orbits for Real Lie Groups Jae-Hyun Yang Department of Mathematics, Inha University, Incheon 402-751, Korea e-mail : [email protected] (2000 Mathematics Subject Classification: Primary 22-XX, 20C35.) In this paper, we outline a development of the theory of orbit method for representa- tions of real Lie groups. In particular, we study the orbit method for representations of the Heisenberg group and the Jacobi group. Contents 1. Introduction 2. Quantization 3. The Kirillov Correspondence 4. Auslander-Kostant’s Theorem 5. The Obstacle for the Orbit Method 5.1. Compact Lie Groups 5.2. Semisimple Lie Groups 6. Nilpotent Orbits and the Kostant-Sekiguchi Correspondence 6.1. Jordan Decomposition 6.2. Nilpotent Orbits 6.3. The Kostant-Sekiguchi Correspondence 6.4. The Quantization of the K-action (due to D. Vogan) 7. Minimal Representations (g,h) 8. The Heisenberg Group HR 8.1. Schr¨odinger Representations 8.2. The Coadjoint Orbits of Picture 9. The Jacobi Group J arXiv:math/0602056v1 [math.RT] 3 Feb 2006 9.1. The Jacobi Group G 9.1.1. The Standard Coordinates of the Jacobi Group GJ 9.1.2. The Iwasawa Decomposition of the Jacobi Group GJ 9.2. The Lie Algebra of the Jacobi Group GJ (Received: December 4, 2001. Revised: May 14, 2002.) Key words and phrases: quantization, the Kirillov correspondence, nilpotent orbits, the Kostant-Sekiguchi correspondence, minimal representations, Heisenberg groups, the Jacobi group. This work was supported by INHA UNIVERSITY Research Grant.(INHA-21382) 199 200 Jae-Hyun Yang 9.3. -
Gravitational Edge Modes, Coadjoint Orbits, and Hydrodynamics
Gravitational Edge Modes, Coadjoint Orbits, and Hydrodynamics William Donnelly,1 Laurent Freidel,1 Seyed Faroogh Moosavian,1;2 Antony J. Speranza1 1Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo ON, N2L 2Y5, Canada 2Department of Physics, McGill University, Ernest Rutherford Physics Building, 3600 Rue Univer- sity, Montr´eal, QC H3A 2T8, Canada E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite- dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner's famous classification of representations of the Poincar´egroup since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincar´egroup. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same arXiv:2012.10367v1 [hep-th] 18 Dec 2020 set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid. -
Coherent States, Holomorphic Extensions, and Highest Weight Representations
Pacific Journal of Mathematics COHERENT STATES, HOLOMORPHIC EXTENSIONS, AND HIGHEST WEIGHT REPRESENTATIONS KARL-HERMANN NEEB Volume 174 No. 2 June 1996 PACIFIC JOURNAL OF MATHEMATICS Vol. 174, No. 2, 1996 COHERENT STATES, HOLOMORPHIC EXTENSIONS, AND HIGHEST WEIGHT REPRESENTATIONS KARL-HERMANN NEEB Let Gbea connected finite dimensional Lie group. In this paper we consider the problem of extending irreducible uni- tary representations of G to holomorphic representations of certain semigroups 5 containing G and a dense open subman- ifold on which the semigroup multiplication is holomorphic. We show that a necessary and sufficient condition for extend- ability is that the unitary representation of G is a highest weight representation. This result provides a direct bridge from representation theory to coadjoint orbits in g*, where g is the Lie algebra of G. Namely the moment map associated naturally to a unitary representation maps the orbit of the highest weight ray (the coherent state orbit) to a coadjoint orbit in g* which has many interesting geometric properties such as certain convexity properties and an invariant complex structure. In this paper we use the interplay between the orbit pic- ture and representation theory to obtain a classification of all irreducible holomorphic representations of the semigroups S mentioned above and a classication of unitary highest weight representations of a rather general class of Lie groups. We also characterize the class of groups and semigroups having sufficiently many highest weight representations to separate the points. 0. Introduction. A closed convex cone W in the Lie algebra Q is called invariant if it is invari- ant under the adjoint action. -
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. -
Causal Symmetric Spaces Geometry and Harmonic Analysis Joachim Hilgert Gestur ´Olafsson
Causal Symmetric Spaces Geometry and Harmonic Analysis Joachim Hilgert Gestur Olafsson´ Contents Preface viii Introduction x 1 Symmetric Spaces 1 1.1 BasicStructureTheory .. .. .. .. .. .. .. .. .. 1 1.2 DualSymmetricSpaces .. .. .. .. .. .. .. .. .. 7 1.2.1 The c-Dual Space ˜ c ................. 7 M 1.2.2 The Associated Dual Space a ........... 9 M 1.2.3 The Riemannian Dual Space r ........... 9 M 1.3 The Module Structure of To(G/H).............. 12 1.4 A-Subspaces........................... 22 1.5 TheHyperboloids........................ 24 2 Causal Orientations 29 2.1 ConvexConesandTheirAutomorphisms . 29 2.2 CausalOrientations . .. .. .. .. .. .. .. .. .. 39 2.3 Semigroups ........................... 43 2.4 TheOrderCompactification. 45 2.5 Examples ............................ 50 2.5.1 TheGroupCase .................... 50 2.5.2 TheHyperboloids . .. .. .. .. .. .. .. .. 51 2.6 Symmetric Spaces Related to Tube Domains . 52 2.6.1 BoundaryOrbits .. .. .. .. .. .. .. .. .. 56 2.6.2 The Functions Ψm ................... 58 2.6.3 The Causal Compactification of .......... 63 M 2.6.4 SU(n,n)......................... 65 2.6.5 Sp(n, R)......................... 68 v vi CONTENTS 3 Irreducible Causal Symmetric Spaces 71 3.1 ExistenceofCausalStructures . 71 3.2 The Classificationof Causal Symmetric Pairs . 83 4 Classification of Invariant Cones 91 4.1 Symmetric SL(2, R)Reduction ................ 91 4.2 TheMinimalandMaximalCones. 98 4.3 TheLinearConvexityTheorem . 105 4.4 TheClassification. 110 4.5 ExtensionofCones. 115 5 The Geometry 120 5.1 The Bounded Realization of H/H K ........... 121 ∩ 5.2 The Semigroup S(C)...................... 126 5.3 TheCausalIntervals . 130 5.4 CompressionSemigroups. 132 5.5 TheNonlinearConvexityTheorem . 143 5.6 The B]-Order.......................... 152 5.7 The Affine Closure of B] .................... 157 6 The Order Compactification 172 6.1 CausalGaloisConnections. -
UNIVERSIT´E DE POITIERS on Representations of Simply
UNIVERSITE´ DE POITIERS On representations of simply connected nilpotent and solvable Lie groups G´erardGr´elaud Universit´ede POITIERS UMR CNRS 6086 – Laboratoire de Math´ematiqueset Applications SP2MI - T´el´eport 2 - Boulevard Marie et Pierre Curie BP 30179 86962 FUTUROSCOPE CHASSENEUIL Cedex T´el: 05 49 49 69 03 e-mail :[email protected] On representations of simply connected nilpotent and solvable Lie groups Introduction In these notes following a course I gave during a visit at Pondicherry University in 1992, I write the main results of the theory of representations of simply connected nilpotent Lie groups (the Kirillov’s theory), and some generalizations to simply connected solvable Lie groups. After the basic properties of unitary representations of locally compact groups, especially construction of induced representations and “Mackey’s machine” (Sec- tion 1), I state the classical results on Lie groups and Lie algebras (section 2). In section 3, I give the constuction of polarizations in solvable Lie algebras and in section 4 the description of the the dual space of connected nilpotent Lie groups, using the famous orbit method of A.A. Kirillov. For general solvable Lie groups, I write in section 5 the construction of irreducible holomorphical induced representations. This shows the use of complex polarizations. The section 6 is devoted to a computation of the Kirillov’s character formula and the Plancherel formula for nilpotent Lie groups and also a generalization to some homogeneous spaces of nilpotent Lie groups. In the last section, I write a survey (and some proofs) of the main results of L. -
Boundary Orbit Strata and Faces of Invariant Cones and Complex Ol’Shanski˘Isemigroups
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 363, Number 7, July 2011, Pages 3799–3828 S 0002-9947(2011)05309-2 Article electronically published on February 15, 2011 BOUNDARY ORBIT STRATA AND FACES OF INVARIANT CONES AND COMPLEX OL’SHANSKI˘ISEMIGROUPS ALEXANDER ALLDRIDGE Abstract. Let D = G/K be an irreducible Hermitian symmetric domain. Then G is contained in a complexification GC, and there exists a closed complex subsemigroup G ⊂ Γ ⊂ GC, the so-called minimal Ol’shanski˘ı semigroup, characterised by the fact that all holomorphic discrete series representations of G extend holomorphically to Γ◦. Parallel to the classical theory of boundary strata for the symmetric domain D, due to Wolf and Kor´anyi, we give a detailed and complete description of the K-orbit type strata of Γ as K-equivariant fibre bundles. They are given by the conjugacy classes of faces of the minimal invariant cone in the Lie algebra. 1. Introduction The boundary structure of Hermitian symmetric domains D = G/K is well understood through the work of Pjatecki˘ı-Shapiro (for the classical domains), and of Wolf and Kor´anyi, in their seminal papers from 1965 [31, 64]: Each of the strata is a K-equivariant fibre bundle whose fibres are Hermitian symmetric domains of lower rank; moreover, the strata are in bijection with maximal parabolic subalgebras. This detailed understanding of the geometry of D has been fruitful and is the basis of a variety of developments in representation theory, harmonic analysis, complex and differential geometry, Lie theory, and operator algebras. Let us mention a few. -
On Hyperbolic Cones and Mixed Symmetric Spaces
Journal of Lie Theory Volume 6 (1996) 69{146 C 1996 Heldermann Verlag On Hyperbolic Cones and Mixed Symmetric Spaces Bernhard Kr¨otz and Karl-Hermann Neeb Communicated by J. Hilgert Abstract. A real semisimple Lie algebra g admits a Cartan involution, θ , for which the corresponding eigenspace decomposition g=k+p has the property that all operators ad X , X p are diagonalizable over . We call 2 R such elements hyperbolic, and the elements X k are elliptic in the sense 2 that ad X is semisimple with purely imaginary eigenvalues. The pairs (g,θ) are examples of symmetric Lie algebras, i.e., Lie algebras endowed with an involutive automorphism, such that the 1 -eigenspace of θ contains − only hyperbolic elements. Let (g,τ) be a symmetric Lie algebra and g= h+q the corresponding eigenspace decomposition for τ . The existence of \enough" hyperbolic elements in q is important for the structural analysis of symmetric Lie algebras in terms of root decompositions with respect to abelian subspaces of q consisting of hyperbolic elements. We study the convexity properties of the action of Inng(h) on the space q . The key role will be played by those invariant convex subsets of q whose interior points are hyperbolic. Introduction It was a fundamental observation of Cartan's that each real semisimple Lie alge- bra g admits an involutive automorphism, nowadays called Cartan involution, θ for which the corresponding eigenspace decomposition g = k+p has the property that all operators ad X , X p are diagonalizable over R, we call such elements hyperbolic, and the elements2X k are elliptic in the sense that ad X is semisim- ple with purely imaginary eigen2values.