808 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
Faces of weight polytopes and a generalization of a theorem of Vinberg
The paper is motivated by the study of graded representations of Takiff
algebras, cominuscule parabolics, and their generalizations. We study certain
special subsets of the set of weights (and of their convex hull) of the
generalized Verma modules (or GVM's) of a semisimple Lie algebra \lie g. In
particular, we extend a result of Vinberg and classify the faces of the convex
hull of the weights of a GVM. When the GVM is finite-dimensional, we ask a
natural question that arises out of Vinberg's result: when are two faces the
same? We also extend the notion of interiors and faces to an arbitrary subfield
\F of the real numbers, and introduce the idea of a weak \F-face of any
subset of Euclidean space. We classify the weak \F-faces of all lattice
polytopes, as well as of the set of lattice points in them. We show that a weak
\F-face of the weights of a finite-dimensional \lie g-module is precisely
the set of weights lying on a face of the convex hull.Comment: Statement changed in Section 4. Typos fixed and some proofs updated.
Submitted to "Algebra and Representation Theory." 18 page
Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups
For the coinvariant rings of finite Coxeter groups of types other than H,
we show that a homogeneous element of degree one is a strong Lefschetz element
if and only if it is not fixed by any reflections. We also give the necessary
and sufficient condition for strong Lefschetz elements in the invariant
subrings of the coinvariant rings of Weyl groups.Comment: 18 page
Su(3) Algebraic Structure of the Cuprate Superconductors Model based on the Analogy with Atomic Nuclei
A cuprate superconductor model based on the analogy with atomic nuclei was
shown by Iachello to have an structure. The mean-field approximation
Hamiltonian can be written as a linear function of the generators of
algebra. Using algebraic method, we derive the eigenvalues of the reduced
Hamiltonian beyond the subalgebras and of
algebra. In particular, by considering the coherence between s- and d-wave
pairs as perturbation, the effects of coherent term upon the energy spectrum
are investigated
Superrigid subgroups and syndetic hulls in solvable Lie groups
This is an expository paper. It is not difficult to see that every group
homomorphism from the additive group Z of integers to the additive group R of
real numbers extends to a homomorphism from R to R. We discuss other examples
of discrete subgroups D of connected Lie groups G, such that the homomorphisms
defined on D can ("virtually") be extended to homomorphisms defined on all of
G. For the case where G is solvable, we give a simple proof that D has this
property if it is Zariski dense. The key ingredient is a result on the
existence of syndetic hulls.Comment: 17 pages. This is the final version that will appear in the volume
"Rigidity in Dynamics and Geometry," edited by M. Burger and A. Iozzi
(Springer, 2002
New synchronization method for <i>Plasmodium falciparum</i>
<b>Background</b>: Plasmodium falciparum is usually asynchronous during in vitro culture. Although various synchronization methods are available, they are not able to narrow the range of ages of parasites. A newly developed method is described that allows synchronization of parasites to produce cultures with an age range as low as 30 minutes.
<b>Methods</b>: Trophozoites and schizonts are enriched using Plasmion. The enriched late stage parasites are immobilized as a monolayer onto plastic Petri dishes using concanavalin A. Uninfected erythrocytes are placed onto the monolayer for a limited time period, during which time schizonts on the monolayer rupture and the released merozoites invade the fresh erythrocytes. The overlay is then taken off into a culture flask, resulting in a highly synchronized population of parasites.
<b>Results</b>: Plasmion treatment results in a 10- to 13-fold enrichment of late stage parasites. The monolayer method results in highly synchronized cultures of parasites where invasion has occurred within a very limited time window, which can be as low as 30 minutes. The method is simple, requiring no specialized equipment and relatively cheap reagents.
<b>Conclusions</b>: The new method for parasite synchronization results in highly synchronized populations of parasites, which will be useful for studies of the parasite asexual cell cycle
Formal Hecke algebras and algebraic oriented cohomology theories
In the present paper we generalize the construction of the nil Hecke ring of
Kostant-Kumar to the context of an arbitrary algebraic oriented cohomology
theory of Levine-Morel and Panin-Smirnov, e.g. to Chow groups, Grothendieck's
K_0, connective K-theory, elliptic cohomology, and algebraic cobordism. The
resulting object, which we call a formal (affine) Demazure algebra, is
parameterized by a one-dimensional commutative formal group law and has the
following important property: specialization to the additive and multiplicative
periodic formal group laws yields completions of the nil Hecke and the 0-Hecke
rings respectively. We also introduce a deformed version of the formal (affine)
Demazure algebra, which we call a formal (affine) Hecke algebra. We show that
the specialization of the formal (affine) Hecke algebra to the additive and
multiplicative periodic formal group laws gives completions of the degenerate
(affine) Hecke algebra and the usual (affine) Hecke algebra respectively. We
show that all formal affine Demazure algebras (and all formal affine Hecke
algebras) become isomorphic over certain coefficient rings, proving an analogue
of a result of Lusztig.Comment: 28 pages. v2: Some results strengthened and references added. v3:
Minor corrections, section numbering changed to match published version. v4:
Sign errors in Proposition 6.8(d) corrected. This version incorporates an
erratum to the published versio
Compact convex sets with prescribed facial dimensions
While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the faces of general closed convex sets. We show that for any finite sequence of positive integers there exist compact convex sets which only have extreme points and faces with dimensions from this prescribed sequence. We also discuss another approach to dimensionality, considering the dimension of the union of all faces of the same dimension. We show that the questions arising from this approach are highly nontrivial and give examples of convex sets for which the sets of extreme points have fractal dimension
Compact convex sets with prescribed facial dimensions
While faces of a polytope form a well structured lattice, in which faces of each possible dimension are present, this is not true for general compact convex sets. We address the question of what dimensional patterns are possible for the faces of general closed convex sets. We show that for any finite sequence of positive integers there exist compact convex sets which only have extreme points and faces with dimensions from this prescribed sequence. We also discuss another approach to dimensionality, considering the dimension of the union of all faces of the same dimension. We show that the questions arising from this approach are highly nontrivial and give examples of convex sets for which the sets of extreme points have fractal dimension
Ethical issues in the use of in-depth interviews: literature review and discussion
This paper reports a literature review on the topic of ethical issues in in-depth interviews. The review returned three
types of article: general discussion, issues in particular studies, and studies of interview-based research ethics. Whilst
many of the issues discussed in these articles are generic to research ethics, such as confidentiality, they often had particular
manifestations in this type of research. For example, privacy was a significant problem as interviews sometimes
probe unexpected areas. For similar reasons, it is difficult to give full information of the nature of a particular interview
at the outset, hence informed consent is problematic. Where a pair is interviewed (such as carer and cared-for) there are
major difficulties in maintaining confidentiality and protecting privacy. The potential for interviews to harm participants
emotionally is noted in some papers, although this is often set against potential therapeutic benefit. As well as
these generic issues, there are some ethical issues fairly specific to in-depth interviews. The problem of dual role is noted
in many papers. It can take many forms: an interviewer might be nurse and researcher, scientist and counsellor, or
reporter and evangelist. There are other specific issues such as taking sides in an interview, and protecting vulnerable
groups. Little specific study of the ethics of in-depth interviews has taken place. However, that which has shows some
important findings. For example, one study shows participants are not averse to discussing painful issues provided they
feel the study is worthwhile. Some papers make recommendations for researchers. One such is that they should consider
using a model of continuous (or process) consent rather than viewing consent as occurring once, at signature, prior
to the interview. However, there is a need for further study of this area, both philosophical and empirical
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