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 It is surely one of the most cited unpublished articles.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 rôle in conformal differential geometry in dimension 4 and is now known as the Paneitz operator

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

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 $(M,g)$ is positive unless $(M,g)$ 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