7,843 research outputs found
Hasse principle for generalised Kummer varieties
The existence of rational points on Kummer varieties associated to
2-coverings of abelian varieties over number fields can sometimes be proved
through the variation of the Selmer group in the family of quadratic twists of
the underlying abelian variety, using an idea of Swinnerton-Dyer. Following
Mazur and Rubin, we consider the case when the Galois action on the 2-torsion
has a large image. Under mild additional hypotheses we prove the Hasse
principle for the associated Kummer varieties assuming the finiteness of
relevant Shafarevich-Tate groups.Comment: 25 page
The orthosymplectic superalgebra in harmonic analysis
We introduce the orthosymplectic superalgebra osp(m|2n) as the algebra of
Killing vector fields on Riemannian superspace R^{m|2n} which stabilize the
origin. The Laplace operator and norm squared on R^{m|2n}, which generate
sl(2), are orthosymplectically invariant, therefore we obtain the Howe dual
pair (osp(m|2n),sl(2)). We study the osp(m|2n)-representation structure of the
kernel of the Laplace operator. This also yields the decomposition of the
supersymmetric tensor powers of the fundamental osp(m|2n)-representation under
the action of sl(2) x osp(m|2n). As a side result we obtain information about
the irreducible osp(m|2n)-representations L_(k,0,...,0). In particular we find
branching rules with respect to osp(m-1|2n) and an interesting formula for the
Cartan product inside the tensor powers of the natural representation of
osp(m|2n). We also prove that integration over the supersphere is uniquely
defined by its orthosymplectic invariance.Comment: partial overlap with arXiv:1202.066
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
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