239 research outputs found

### The Minkowski and conformal superspaces

We define complex Minkowski superspace in 4 dimensions as the big cell inside
a complex flag supermanifold. The complex conformal supergroup acts naturally
on this super flag, allowing us to interpret it as the conformal
compactification of complex Minkowski superspace. We then consider real
Minkowski superspace as a suitable real form of the complex version. Our
methods are group theoretic, based on the real conformal supergroup and its
Lie superalgebra.Comment: AMS LaTeX, 44 page

### Dieudonn\'e modules and $p$-divisible groups associated with Morava $K$-theory of Eilenberg-Mac Lane spaces

We study the structure of the formal groups associated to the Morava
$K$-theories of integral Eilenberg-Mac Lane spaces. The main result is that
every formal group in the collection $\{K(n)^*K({\mathbb Z}, q), q=2,3,...\}$
for a fixed $n$ enters in it together with its Serre dual, an analogue of a
principal polarization on an abelian variety. We also identify the isogeny
class of each of these formal groups over an algebraically closed field. These
results are obtained with the help of the Dieudonn\'e correspondence between
bicommutative Hopf algebras and Dieudonn\'e modules. We extend P. Goerss's
results on the bilinear products of such Hopf algebras and corresponding
Dieudonn\'e modules.Comment: 23 page

### Extention of Finite Solvable Torsors over a Curve

Let $R$ be a discrete valuation ring with fraction field $K$ and with
algebraically closed residue field of positive characteristic $p$. Let $X$ be a
smooth fibered surface over $R$ with geometrically connected fibers endowed
with a section $x\in X(R)$. Let $G$ be a finite solvable $K$-group scheme and
assume that either $|G|=p^n$ or $G$ has a normal series of length 2. We prove
that every quotient pointed $G$-torsor over the generic fiber $X_{\eta}$ of $X$
can be extended to a torsor over $X$ after eventually extending scalars and
after eventually blowing up $X$ at a closed subscheme of its special fiber
$X_s$.Comment: 16 page

### On character generators for simple Lie algebras

We study character generating functions (character generators) of simple Lie
algebras. The expression due to Patera and Sharp, derived from the Weyl
character formula, is first reviewed. A new general formula is then found. It
makes clear the distinct roles of ``outside'' and ``inside'' elements of the
integrity basis, and helps determine their quadratic incompatibilities. We
review, analyze and extend the results obtained by Gaskell using the Demazure
character formulas. We find that the fundamental generalized-poset graphs
underlying the character generators can be deduced from such calculations.
These graphs, introduced by Baclawski and Towber, can be simplified for the
purposes of constructing the character generator. The generating functions can
be written easily using the simplified versions, and associated Demazure
expressions. The rank-two algebras are treated in detail, but we believe our
results are indicative of those for general simple Lie algebras.Comment: 50 pages, 11 figure

### Restricted infinitesimal deformations of restricted simple Lie algebras

We compute the restricted infinitesimal deformations of the restricted simple
Lie algebras over an algebraically closed field of characteristic different
from 2 and 3.Comment: 15 pages; final version, to appear in Journal of Algebra and Its
Application

### The Optimal Control Landscape for the Generation of Unitary Transformations with Constrained Dynamics

The reliable and precise generation of quantum unitary transformations is
essential to the realization of a number of fundamental objectives, such as
quantum control and quantum information processing. Prior work has explored the
optimal control problem of generating such unitary transformations as a surface
optimization problem over the quantum control landscape, defined as a metric
for realizing a desired unitary transformation as a function of the control
variables. It was found that under the assumption of non-dissipative and
controllable dynamics, the landscape topology is trap-free, implying that any
reasonable optimization heuristic should be able to identify globally optimal
solutions. The present work is a control landscape analysis incorporating
specific constraints in the Hamiltonian corresponding to certain dynamical
symmetries in the underlying physical system. It is found that the presence of
such symmetries does not destroy the trap-free topology. These findings expand
the class of quantum dynamical systems on which control problems are
intrinsically amenable to solution by optimal control.Comment: Submitted to Journal of Mathematical Physic

### A coproduct structure on the formal affine Demazure algebra

In the present paper we generalize the coproduct structure on nil Hecke rings
introduced and studied by Kostant-Kumar to the context of an arbitrary
algebraic oriented cohomology theory and its associated formal group law. We
then construct an algebraic model of the T-equivariant oriented cohomology of
the variety of complete flags.Comment: 28 pages; minor revision of the previous versio

### A differential method for bounding the ground state energy

For a wide class of Hamiltonians, a novel method to obtain lower and upper
bounds for the lowest energy is presented. Unlike perturbative or variational
techniques, this method does not involve the computation of any integral (a
normalisation factor or a matrix element). It just requires the determination
of the absolute minimum and maximum in the whole configuration space of the
local energy associated with a normalisable trial function (the calculation of
the norm is not needed). After a general introduction, the method is applied to
three non-integrable systems: the asymmetric annular billiard, the many-body
spinless Coulombian problem, the hydrogen atom in a constant and uniform
magnetic field. Being more sensitive than the variational methods to any local
perturbation of the trial function, this method can used to systematically
improve the energy bounds with a local skilled analysis; an algorithm relying
on this method can therefore be constructed and an explicit example for a
one-dimensional problem is given.Comment: Accepted for publication in Journal of Physics

### Cohomology of the minimal nilpotent orbit

We compute the integral cohomology of the minimal non-trivial nilpotent orbit
in a complex simple (or quasi-simple) Lie algebra. We find by a uniform
approach that the middle cohomology group is isomorphic to the fundamental
group of the sub-root system generated by the long simple roots. The modulo
$\ell$ reduction of the Springer correspondent representation involves the sign
representation exactly when $\ell$ divides the order of this cohomology group.
The primes dividing the torsion of the rest of the cohomology are bad primes.Comment: 29 pages, v2 : Leray-Serre spectral sequence replaced by Gysin
sequence only, corrected typo

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