Deformation Quantization on the Closure of Minimal Coadjoint Orbits

We consider a complex simple Lie algebra ${\mathfrak{g}}$ , with the action of its adjoint group. Among the three canonical nilpotent orbits under this action, the minimal orbit is the non zero orbit of smallest dimension. We are interested in equivariant deformation quantization: we construct ${\mathfrak{g}}$ -invariant star-products on the minimal orbit and on its closure, a singular algebraic variety. We shall make use of Hochschild homology and cohomology, of some results about the invariants of the classical groups, and of some interesting representations of simple Lie algebras. To the minimal orbit is associated a unique, completely prime two-sided ideal of the universal enveloping algebra ${{\rm U}(\mathfrak{g})}$ . This ideal is primitive and is called the Joseph ideal. We give explicit expressions for the generators of the Joseph ideal and compute the infinitesimal characters.     

Darboux Coordinates on Coadjoint Orbits of Lie Algebras

The method of constructing spectral Darboux coordinates on finite-dimensional coadjoint orbits induals of loop algebras is applied to the one pole case, where the orbit isidentified with a coadjoint orbit in the dual of a finite-dimensional Liealgebra. The constructions are carried out explicitly when the Lie algebra is sl(2, R),sl(3,R), andso(3, R), and for rank twoorbits is so(n, R). A new feature thatappears is the possibility of identifying spectralDarboux coordinatesassociated to dynamical choices of sections of the associatedeigenvector line bundles; i.e.sections that depend on the point within the given orbit.     

On Quantizing Nonnilpotent Coadjoint Orbits of Semisimple Lie Groups

We prove that there is no consistent polynomial quantization of the coordinate ring of a nonnilpotent coadjoint orbit of a semisimple Lie group.     

Coadjoint Orbits of Certain Motion Groups and Their Coherent States

Let ?_n, n ≥ 1, be the (2n+1)-dimensional Heisenberg group and let K be a closed connected subgroup of the unitary group U(n) acting on ?_n by automorphisms. Using the moment map, we provide in this paper a dequantization procedure for all generic admissible coadjoint orbits of the semidirect product G = K ? ?_n. In the opposite direction, we show that Gilmore-Perelomov's coherent states define “pure state quantizations” of such orbits.     

Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups

Let ? be the function algebra on a semisimple orbit, M, in the coadjoint representation of a simple Lie group, g, with the Lie algebra ?. We study one and two parameter quantizations ? _h and ? _t,h of ? such that the multiplication on the quantized algebra is invariant under action of the Drinfeld–Jimbo quantum group, U _{h (?)}. In particular, the algebra ? _t,h specializes at h= 0 to a U(?)-invariant ($G$-invariant) quantization, %Ascr; _{t ,0}. We prove that the Poisson bracket corresponding to ? _h must be the sum of the so-called r-matrix and an invariant bracket. We classify such brackets for all semisimple orbits, M, and show that they form a dim H ²(M) parameter family, then we construct their quantizations. A two parameter (or double) quantization, $? _t,h , corresponds to a pair of compatible Poisson brackets: the first is as described above and the second is the Kirillov-Kostant-Souriau bracket on M. Not all semisimple orbits admit a compatible pair of Poisson brackets. We classify the semisimple orbits for which such pairs exist and construct the corresponding two parameter quantization of these pairs in some of the cases. Received: 15 August 1998 / Accepted: 13 January 1999     

Explicit Equivariant Quantization on Coadjoint Orbits of GL(n,C)

We present an explicit U _h (gl(n, C))-equivariant quantization on coadjoint orbits of GL(n, C). It forms a two-parameter family quantizing the Poisson pair of the reflection equation and Kirillov–Kostant–Souriau brackets.     

A Weyl group for the Virasoro andN=1 super-Virasoro algebras

We introduce a Weyl group for the highest weight modules over the Virasoro algebra and the Neveu-Schwarz and Ramond superalgebras. Using this group we rewrite the character formulae for the irreducible highest weight modules over these algebras in the form of the classical Weyl character formula for the finite-dimensional irreducible representations of semi-simple Lie algebras (and also of the Weyl-Kac character formula for the integrable highest weight modules over affine Kac-Moody algebras). This is the same group we introduced recently in order to rewrite in a similar manner the characters of the singular highest weight modules over the affine Kac-Moody algebraA ₁ ⁽¹⁾ .     

Real Forms of the Complex Twisted N=2 Supersymmetric Toda Chain Hierarchy in Real N=1 and Twisted N=2 Superspaces

Abstract

Three nonequivalent real forms of the complex twisted N=2 supersymmetric Toda chain hierarchy (solv-int/9907021) in real N=1 superspace are presented. It is demonstrated that they possess a global twisted N=2 supersymmetry. We discuss a new superfield basis in which the supersymmetry transformations are local. Furthermore, a representation of this hierarchy is given in terms of two twisted chiral N=2 superfields. The relations to the s-Toda hierarchy by H. Aratyn, E. Nissimov and S. Pacheva (solv-int/9801021) as well as to the modified and derivative NLS hierarchies are established.     

3维N=1超引力理论和超Higgs效应

本文利用超Poincare张量运算,构造了一个3维N=1超引力理论,给出了一般的拉氏密度,详细讨论了超对称自发破缺机制及超Higgs效应.     

Extended supergravity on the supergroup manifold: N = 3 and N = 2 theories

In this paper I construct the group-manifold first-order formulation of N = 2 and N = 3 supergravity based on the $\text{Osp}\text{(}\text{4}\text{2}\text{)}\text{and Osp}\text{(}\text{4}\text{3}\text{)}$ supergroups, respectively. In the case N = 2, a group manifold version of the theory was already presented in a previous paper. Here a simpler formulation is given which shows exact factorization in the SO(2) subgroup absent in the previous one. Particular attention is paid to the algebraic role played by the $\text{spin}\text{-}\text{1}\text{2}$ field which is the novel feature of the N = 3 case with respect to N = 2. It is shown how the “non-geometrical” term in the gravitino transformation law in the N = 2 theory arises from the rheonomic symmetry mechanism.     

Hamiltonian N=2 superfield quantization

We present a superfield construction of Hamiltonian quantization with N=2 supersymmetry generated by two fermionic charges Q^a. As a byproduct of the analysis we also derive a classically localized path integral from two fermionic objects Σ^a that can be viewed as “square roots” of the classical bosonic action under the product of a functional Poisson bracket.