Adjoint Orbits Research Articles

Special Transverse Slices and Their Enveloping Algebras

Let G be a simple, simply connected algebraic group over C , g =Lie G, N ( g ) the nilpotent cone of g , and ( E, H, F) an s l 2-triple in g . Let S= E+Ker ad F, the special transverse slice to the adjoint orbit Ω of E, and S 0= S∩ N ( g ). The coordinate ring C [ S 0] is naturally graded (See Slodowy, “Simple Singularities and Simple Algebraic Groups,” Lecture Notes in Mathematics, Vol. 815, Springer-Verlag, Berlin/Heidelberg/New York, 1980). Let Z( g ) be the centre of the enveloping algebra U( g ) and η: Z( g )→ C an algebra homomorphism. Identify g with g * via a Killing isomorphism and let χ denote the linear function on g corresponding to E. Following Kawanaka (Generalized Gelfand–Graev representations and Ennola duality, in “Algebraic Groups and Related Topics” Advanced Studies in Pure Mathematics, Vol. 6, pp. 175–206, North-Holland, Amsterdam/New York/Oxford, 1985), Moeglin (C.R. Acad. Sci. Paris, Ser. I 303 No. 17 (1986), 845–848), and Premet ( Invent. Math. 121 (1995), 79–117), we attach to χ a nilpotent subalgebra m χ ⊂ g of dimension (dim Ω)/2 and a 1-dimensional m χ -module C χ . Let H̃ χ denote the algebra opposite to End g ( U( g )⊗ U( m χ) C χ ) and H̃ χ,η = H̃ χ ⊗ Z( g ) C η . It is proved in the paper that the algebra H̃ χ,η has a natural filtration such that gr( H̃ χ,η ), the associated graded algebra, is isomorphic to C [ S 0]. This construction yields natural noncommutative deformations of all singularities associated with the adjoint quotient map of g .

GEOMETRY OF MIXED STATES AND DEGENERACY STRUCTURE OF GEOMETRIC PHASES FOR MULTI-LEVEL QUANTUM SYSTEMS: A UNITARY GROUP APPROACH

We analyze the geometric aspects of unitary evolution of general states for a multilevel quantum system by exploiting the structure of coadjoint orbits in the unitary group Lie algebra. Using the same methods in the case of SU(3) we study the effect of degeneracies on geometric phases for three-level systems. This is shown to lead to a highly nontrivial generalization of the result for two-level systems in which degeneracy results in a "monopole" structure in parameter space. The rich structures that arise are related to the geometry of adjoint orbits in SU(3). The limiting case of a two-level degeneracy in a three-level system is shown to lead to the known monopole structure.

Completely integrable curve flows on Adjoint orbits

It is known that the Schrödinger flow on a complex Grassmann manifold is equivalent to the matrix non-linear Schrödinger equation and the Ferapontov flow on a principal Adjoint U(n)-orbit is equivalent to the n-wave equation. In this paper, we give a systematic method to construct integrable geometric curve flows on Adjoint U-orbits from flows in the soliton hierarchy associated to a compact Lie group U. There are natural geometric bi-Hamiltonian structures on the space of curves on Adjoint orbits, and they correspond to the order two and three Hamiltonian structures on soliton equations under our construction. We study the Hamiltonian theory of these geometric curve flows and also give several explicit examples.

Sheets and Topology of Primitive Spectra for Semisimple Lie Algebras

Let g be a semisimple Lie algebra. Consider the set X of primitive ideals of the enveloping algebra U(g), given the Jacobson topology. A basic open problem is to describe X as a countable union of algebraic varieties V with strata V as large as possible. Towards this aim the notion of a sheet in X is introduced here in analogy with the notion of sheet in the adjoint orbit space g/G. The sheets in X are classified and given a purely topological description. Moreover Goldie-rank is constant and maximal on a dense open subset of a sheet. For every positive integer n it is shown that there are only finitely many sheets in X with maximal Goldie-rank n. For n=1, which corresponds to completely prime ideals, an extension of the Dixmier-map (from some adjoint orbits to primitive ideals) is introduced with the aim of showing that the sheets with n=1 are homeomorphic to sheets in g/G and hence explicitly described as algebraic varieties. This goal is partly though not fully achieved. The proofs of the above results require a detailed knowledge of Goldie-rank polynomials and even a new difficult result concerning the support of their restrictions to the walls. An extension of a theorem of Soergel on the topology of X is also required, and for this a new approach is given, which is both simpler and more general.

Isospectral gradient flows for non-symmetric eigenvalue problem

We derive the general formula of isospectral gradient flows on adjoint orbits which are related to the theory of solitons and matrix eigenvalue problem. The general formula includes some known examples of isospectral flows. Using the general formula, we obtain new dynamical systems which provide new computational algorithm for solving the eigenvalue problem of non-symmetric matrices.

B-Orbits in Solutions to the Equation X2 = 0 in Triangular Matrices

Consider In(K), the set of solutions to the equation X2=0 in n×n strictly upper-triangular matrices over field K. In this paper we construct the bijection from the set of adjoint orbits in In(K) onto the set of involutions in symmetric group Sn. For a finite field K we compute the number of points in the orbits.

QUATERNIONIC KAHLER MANIFOLDS OF COHOMOGENEITY ONE

Classification results are given for (i) compact quaternionic Kähler manifolds with a cohomogeneity-one action of a semi-simple group, (ii) certain complete hyperKähler manifolds with a cohomogeneity-two action of a semi-simple group preserving each complex structure, (iii) compact 3-Sasakian manifolds which are cohomogeneity one with respect to a group of 3-Sasakian symmetries. Information is also obtained about non-compact quaternionic Kähler manifolds of cohomogeneity one and the cohomogeneity of adjoint orbits in complex semi-simple Lie algebras.

Symplectic singularities

We discuss a particular class of rational Gorenstein singularities, which we call symplectic. A normal variety V has symplectic singularities if its smooth part carries a closed symplectic 2-form whose pull-back in any resolution X --> V extends to a holomorphic 2-form on X . Our main result is the classification of isolated symplectic singularities with smooth projective tangent cone. They are in one-to-one correspondence with simple complex Lie algebras: to a Lie algebra g corresponds the singularity at 0 of the closure of the minimal (nonzero) nilpotent adjoint orbit in g .

Hopf term, fractional spin and soliton operators in the O(3) non-linear sigma model

We re-examine three issues, the Hopf term, fractional spin and the soliton operators, in the 2+1 dimensional O(3) non-linear sigma model based on the adjoint orbit parametrization (AOP) introduced earlier. It is shown that the Hopf term is well defined for configurations of any soliton charge Q if we adopt a time-independent boundary condition at spatial infinity. We then develop the Hamiltonian formulation of the model in the AOP and thereby argue that the well-known Q 2 formula for fractional spin holds only for a restricted class of configurations. Operators that create states of given classical configurations of any soliton number in the (physical) Hilbert space are constructed. Our results clarify some of the points that are crucial for the above three topological issues and yet have remained obscure in the literature.

Explicit Cayley triples in real forms of G2, F4, and E6

We re-examine the problem of classifying the nilpotent adjoint orbits in semisimple real Lie algebras. We present the list of representatives of these orbits, as well as those of related Cayley triples, in the case when the complexication of the algebra is one of the exceptional complex Lie algebras mentioned in the title.

Quantum mechanically induced Hopf term in the O(3) nonlinear sigma model

The Hopf term in the (2+ 1)-dimensional O(3) nonlinear sigma model, which is known to be responsible for fractional spin and statistics, is re-examined from the viewpoint of quantization ambiguity. It is confirmed that the Hopf term can be understood as a term induced quantum mechanically, in precisely the same manner as the θ-term in QCD. We present a detailed analysis of the topological aspect of the model based on the adjoint orbit parametrization of the spin vectors, which is not only very useful in handling topological (soliton and/or Hopf) numbers, but also plays a crucial role in defining the Hopf term for configurations of nonvanishing soliton numbers. The Hopf term is seen to arise explicitly as a quantum effect which emerges when quantizing an S 1 degree of freedom hidden in the configuration space.

Quantization of the 4-Dimensional Nilpotent Orbit of Sl(3, ℝ)

AbstractWe give a new geometric model for the quantization of the 4-dimensional conical (nilpotent) adjoint orbit Oℝof SL(3,ℝ). The space of quantization is the space of holomorphic functions on 𝕔2- {0}) which are square integrable with respect to a signed measure defined by a Meijer G-function. We construct the quantization out a non-flat Kaehler structure on 𝕔2- {0}) (the universal cover of Oℝ) with Kaehler potential ρ |z|4.

Twistor Geometry for Hyperkähler Metrics on Complex Adjoint Orbits

We study the hyperkahler geometry of complex adjoint orbits from the point of view of twistor theory. We introduce, for complex semisimple adjoint orbits, the associated spectral curve and construct the twistor space as a union of certain regular adjoint orbits; we also exhibit the family of twistor lines. Furthermore, we show how our methods may be applied for describing hyperkahler metrics associated to more general spectral curves. In particular, we give an algebraic characterisation of the twistor lines.

Finite Fields, Root Systems, and Orbit Numbers of Chevalley Groups

We describe combinatorial techniques to determine the numbers of semisimple conjugacy classes and adjoint orbits with fixed class of centralizers for simply connected finite groups of Lie type.

The Floer homotopy type of height functions on complex Grassmann manifolds

A family of Floer functions on the infinite dimensional complex Grassmann manifold is defined by taking direct limits of height functions on adjoint orbits of unitary groups. The Floer cohomology of a generic function in the family is computed using the Schubert calculus. The Floer homotopy type of this function is computed and the Floer cohomology which was computed algebraically is recovered from the Floer homotopy type. Certain non-generic elements of this family of Floer functions were shown to be related to the symplectic action functional on the universal cover of the loop space of a finite dimensional complex Grassmann manifold in the author’s preprint The Floer homotopy type of complex Grassmann manifolds.

Hyper-lie poisson structures

The main purpose of the paper is to study hyperkähler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkähler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras [10], [11], we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra and give criterion which implies their existence. We study an explicit example of a hyper-Lie Poisson structure, in which the moduli spaces of solutions to Nahm's equations assocaited to Lie algebra su(2) are realized as hypersymplectic leaves and are related to the (co)adjoint orbits of sl(2,).

Optimal control and geodesic flows

In this paper we analyze and generalize, from the point of view of the maximum principle, a class of nonlinear optimal control problems originally introduced in Brockett (1994). The optimal control problems discussed here lie on the adjoint orbits of compact Lie groups or on Grassmann manifolds and the structure of the equations that arise depends on the metric used - the so-called normal metric. We consider some special cases of the general problem and their meaning from the point of view of Riemannian geometry and Hamiltonian mechanics.

Local Character Expansions for Supercuspidal Representations of U(3)

AbstractThe topic of this paper is the relationship between characters of irreducible supercuspidal representations of the p-adic unramified 3 x 3 unitary group and Fourier transforms of invariant measures on elliptic adjoint orbits in the Lie algebra. We prove that most supercuspidal representations have the property that, on some neighbourhood of zero, the character composed with the exponential map coincides with the formal degree of the representation times the Fourier transform of a measure on one elliptic orbit. For the remainder, a linear combination of the Fourier transforms of measures on two elliptic orbits must be taken. As a consequence of these relations between characters and Fourier transforms, the coefficients in the local character expansions are expressed in terms of values of Shalika germs. By calculating which of the values of the Shalika germs associated to regular nilpotent orbits are nonzero, we determine which irreducible supercuspidal representations have Whittaker models. Finally, the coefficients in the local character expansions of three families of supercuspidal representations are computed.

Quantum states of string-inspired lineal gravity.

We construct quantum states for a (1+1) dimensional gravity-matter model that is also a gauge theory based on the centrally extended Poincar\'e group. Explicit formulas are found, which exhibit interesting structures. For example wave functionals are gauge invariant except for a gauge non-invariant phase factor that is the Kirillov-Kostant 1-form on the (co-) adjoint orbit of the group. However no evidence for gravity-matter forces is found.

A Schur-Horn-Kostant convexity theorem for the diffeomorphism group of the annulus

The group of area preserving diffeomorphisms of the annulus acts on its Lie algebra, the globally Hamiltonian vectorfields on the annulus. We consider a certain Hilbert space completion of this group (thinking of it as a group of unitary operators induced by the diffeomorphisms), and prove that the projection of an adjoint orbit onto a “Cartan” subalgebra isomorphic toL2 ([0, 1]) is an infinite-dimensional, weakly compact, convex set, whose extreme points coincide with the orbit, through a certain function, of the “permutation” semigroup of measure preserving transformations of [0, 1].