117 research outputs found
Indecomposable finite-dimensional representations of a class of Lie algebras and Lie superalgebras
In the article at hand, we sketch how, by utilizing nilpotency to its fullest
extent (Engel, Super Engel) while using methods from the theory of universal
enveloping algebras, a complete description of the indecomposable
representations may be reached. In practice, the combinatorics is still
formidable, though.
It turns out that the method applies to both a class of ordinary Lie algebras
and to a similar class of Lie superalgebras.
Besides some examples, due to the level of complexity we will only describe a
few precise results. One of these is a complete classification of which ideals
can occur in the enveloping algebra of the translation subgroup of the
Poincar\'e group. Equivalently, this determines all indecomposable
representations with a single, 1-dimensional source. Another result is the
construction of an infinite-dimensional family of inequivalent representations
already in dimension 12. This is much lower than the 24-dimensional
representations which were thought to be the lowest possible. The complexity
increases considerably, though yet in a manageable fashion, in the
supersymmetric setting. Besides a few examples, only a subclass of ideals of
the enveloping algebra of the super Poincar\'e algebra will be determined in
the present article.Comment: LaTeX 14 page
A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
This is the original manuscript dated March 9th 1983, typeset by the Editors
for the Proceedings of the Midwest Geometry Conference 2007 held in memory of
Thomas Branson. Stephen Paneitz passed away on September 1st 1983 while
attending a conference in Clausthal and the manuscript was never published. For
more than 20 years these few pages were circulated informally. In November
2004, as a service to the mathematical community, Tom Branson added a scan of
the manuscript to his website. Here we make it available more formally. It is
surely one of the most cited unpublished articles. The differential operator
defined in this article plays a key role in conformal differential geometry in
dimension 4 and is now known as the Paneitz operator.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
The Yang-Mills equations on the universal cosmos
AbstractGlobal existence and regularity of solutions for the Yang-Mills equations on the universal cosmos M̃, which has the form R1 × S3 for each of an 8-parameter continuum of factorizations of M̃ as time × space, are treated by general methods. The Cauchy problem in the temporal gauge is globally soluble in its abstract evolutionary form with arbitrary data for the field ⊕ potential in L2,r(S3) ⊕ L2,r + 1(S3), where r is an integer >1 and L2,r denotes the class of sections whose first r derivatives are square-integrable; if r = 1, the problem is soluble locally in time. When r is 3 or more the solution is identifiable with a classical one; if infinite, the solution is in C∞(M̃). These results extend earlier work and approaches [1–5]. Solutions of the equations on Minkowski space-time M0 extend canonically (modulo gauge transformations) to solutions on M̃ provided their Cauchy data are moderately smooth and small near spatial infinity. Precise asymptotic structures for solutions on M0 follow, and in turn imply various decay estimates. Thus the energy in regions uniformly bounded in direction away from the light cone is O(¦x0¦−5), where x0 is the Minkowski time coordinate; analysis solely in M0 [8,9] earlier yielded the estimate O(¦x0¦−2) applicable to the region within the light cone. Similarly it follows that the action integral for a solution of the Yang-Mills equations in M0 is finite, in fact absolutely convergent
Singular operators on boson fields as forms on spaces of entire functions on Hilbert space
AbstractInvariant scales of entire analytic functions on Hilbert space are introduced and applied. Singular operators represented by sesquilinear forms on spaces of regular vectors are given explicit integral representations via kernels that are entire functions on the direct sum of the Hilbert space with its dual. The Weyl (or, exponentiated boson field) operators act smoothly and irreducibly on corresponding spaces of entire functions. Arbitrary symplectic operators on a single-particle Hilbert space are shown to be implementable on the corresponding boson field by appropriate generalized operators
A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
This is the original manuscript dated March 9th 1983, typeset by the Editors for the Proceedings of the Midwest Geometry Conference 2007 held in memory of Thomas Branson. Stephen Paneitz passed away on September 1st 1983 while attending a conference in Clausthal and the manuscript was never published. For more than 20 years these few pages were circulated informally. In November 2004, as a service to the mathematical community, Tom Branson added a scan of the manuscript to his website. Here we make it available more formally. It is surely one of the most cited unpublished articles. The differential operator defined in this article plays a key rôle in conformal differential geometry in dimension 4 and is now known as the Paneitz operator
Positive mass theorem for the Paneitz-Branson operator
We prove that under suitable assumptions, the constant term in the Green
function of the Paneitz-Branson operator on a compact Riemannian manifold
is positive unless is conformally diffeomophic to the standard
sphere. The proof is inspired by the positive mass theorem on spin manifolds by
Ammann-Humbert.Comment: 7 page
A New Family of Gauges in Linearized General Relativity
For vacuum Maxwell theory in four dimensions, a supplementary condition
exists (due to Eastwood and Singer) which is invariant under conformal
rescalings of the metric, in agreement with the conformal symmetry of the
Maxwell equations. Thus, starting from the de Donder gauge, which is not
conformally invariant but is the gravitational counterpart of the Lorenz gauge,
one can consider, led by formal analogy, a new family of gauges in general
relativity, which involve fifth-order covariant derivatives of metric
perturbations. The admissibility of such gauges in the classical theory is
first proven in the cases of linearized theory about flat Euclidean space or
flat Minkowski space-time. In the former, the general solution of the equation
for the fulfillment of the gauge condition after infinitesimal diffeomorphisms
involves a 3-harmonic 1-form and an inverse Fourier transform. In the latter,
one needs instead the kernel of powers of the wave operator, and a contour
integral. The analysis is also used to put restrictions on the dimensionless
parameter occurring in the DeWitt supermetric, while the proof of admissibility
is generalized to a suitable class of curved Riemannian backgrounds.
Eventually, a non-local construction is obtained of the tensor field which
makes it possible to achieve conformal invariance of the above gauges.Comment: 28 pages, plain Tex. In the revised version, sections 4 and 5 are
completely ne
Conformally invariant bending energy for hypersurfaces
The most general conformally invariant bending energy of a closed
four-dimensional surface, polynomial in the extrinsic curvature and its
derivatives, is constructed. This invariance manifests itself as a set of
constraints on the corresponding stress tensor. If the topology is fixed, there
are three independent polynomial invariants: two of these are the
straighforward quartic analogues of the quadratic Willmore energy for a
two-dimensional surface; one is intrinsic (the Weyl invariant), the other
extrinsic; the third invariant involves a sum of a quadratic in gradients of
the extrinsic curvature -- which is not itself invariant -- and a quartic in
the curvature. The four-dimensional energy quadratic in extrinsic curvature
plays a central role in this construction.Comment: 16 page
Singular limits for the bi-laplacian operator with exponential nonlinearity in
Let be a bounded smooth domain in such that for
some integer its -th singular cohomology group with coefficients in
some field is not zero, then problem
{\Delta^{2}u-\rho^{4}k(x)e^{u}=0 & \hbox{in}\Omega,
u=\Delta u=0 & \hbox{on}\partial\Omega,
has a solution blowing-up, as , at points of , for any
given number .Comment: 30 pages, to appear in Ann. IHP Non Linear Analysi
Vacuum polarization in Schwarzschild space-time by anomaly induced effective actions
The characteristic features of in the Boulware, Unruh and
Hartle-Hawking states for a conformal massless scalar field propagating in the
Schwarzschild space-time are obtained by means of effective actions deduced by
the trace anomaly. The actions are made local by the introduction of auxiliary
fields and boundary conditions are carefully imposed on them in order to select
the different quantum states.Comment: 20 pages, latex; misprints corrected and references adde
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