Rolling of Coxeter polyhedra along mirrors

The topic of the paper are developments of $n$-dimensional Coxeter polyhedra. We show that the surface of such polyhedron admits a canonical cutting such that each piece can be covered by a Coxeter $(n-1)$-dimensional domain.Comment: 20pages, 15 figure

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

Topics on global analysis of manifolds and representation theory of reductive groups

Geometric symmetry induces symmetries of function spaces, and the latter yields a clue to global analysis via representation theory. In this note we summarize recent developments on the general theory about how geometric conditions affect representation theoretic properties on function spaces, with focus on multiplicities and spectrum.Comment: 14 page

Counting Exceptional Instantons

We show how to obtain the instanton partition function of N=2 SYM with exceptional gauge group EFG using blow-up recursion relations derived by Nakajima and Yoshioka. We compute the two instanton contribution and match it with the recent proposal for the superconformal index of rank 2 SCFTs with E6, E7 global symmetry.Comment: 16 pages, references adde

Supersymmetric Deformations of Maximally Supersymmetric Gauge Theories

We study supersymmetric and super Poincar\'e invariant deformations of ten-dimensional super Yang-Mills theory and of its dimensional reductions. We describe all infinitesimal super Poincar\'e invariant deformations of equations of motion of ten-dimensional super Yang-Mills theory and its reduction to a point; we discuss the extension of them to formal deformations. Our methods are based on homological algebra, in particular, on the theory of L-infinity and A-infinity algebras. The exposition of this theory as well as of some basic facts about Lie algebra homology and Hochschild homology is given in appendices.Comment: New results added. 111 page

Hilbert Series for Moduli Spaces of Two Instantons

The Hilbert Series (HS) of the moduli space of two G instantons on C^2, where G is a simple gauge group, is studied in detail. For a given G, the moduli space is a singular hyperKahler cone with a symmetry group U(2) \times G, where U(2) is the natural symmetry group of C^2. Holomorphic functions on the moduli space transform in irreducible representations of the symmetry group and hence the Hilbert series admits a character expansion. For cases that G is a classical group (of type A, B, C, or D), there is an ADHM construction which allows us to compute the HS explicitly using a contour integral. For cases that G is of E-type, recent index results allow for an explicit computation of the HS. The character expansion can be expressed as an infinite sum which lives on a Cartesian lattice that is generated by a small number of representations. This structure persists for all G and allows for an explicit expressions of the HS to all simple groups. For cases that G is of type G_2 or F_4, discrete symmetries are enough to evaluate the HS exactly, even though neither ADHM construction nor index is known for these cases.Comment: 53 pages, 9 tables, 24 figure

On extensions of some classes of algebras

The paper consists of three parts. In the first part we discuss on extensions of Lie algebras and their importance in Physics. Then we deal with the extensions of some classes of algebras with one binary operation. The third part is devoted to the study of extensions of two classes of algebras, possessing two algebraic operations, called dialgebras. In all the cases we propose 2-cocycles and respective extensions. The motivation to study the extensions is to use them further for the classification problem of the classes algebras considered in low dimensional cases

Rationale and design of the participant, investigator, observer, and data-analyst-blinded randomized AGENDA trial on associations between gene-polymorphisms, endophenotypes for depression and antidepressive intervention: the effect of escitalopram versus placebo on the combined dexamethasone-corticotrophine releasing hormone test and other potential endophenotypes in healthy first-degree relatives of persons with depression

<p>Abstract</p> <p>Background</p> <p>Endophenotypes are heritable markers, which are more prevalent in patients and their healthy relatives than in the general population. Recent studies point at disturbed regulation of the hypothalamic-pituitary-adrenocortical axis as a possible endophenotype for depression. We hypothesize that potential endophenotypes for depression may be affected by selective serotonin re-uptake inhibitor antidepressants in healthy first-degree relatives of depressed patients. The primary outcome measure is the change in plasma cortisol in the dexamethasone-corticotrophin releasing hormone test from baseline to the end of intervention.</p> <p>Methods</p> <p>The AGENDA trial is designed as a participant, investigator, observer, and data-analyst-blinded randomized trial. Participants are 80 healthy first-degree relatives of patients with depression. Participants are randomized to escitalopram 10 mg per day versus placebo for four weeks. Randomization is stratified by gender and age. The primary outcome measure is the change in plasma cortisol in the dexamethasone-corticotrophin releasing hormone test at entry before intervention to after four weeks of intervention. With the inclusion of 80 participants, a 60% power is obtained to detect a clinically relevant difference in the primary outcome between the intervention and the placebo group. Secondary outcome measures are changes from baseline to four weeks in scores of: 1) cognition and 2) neuroticism. Tertiary outcomes measures are changes from baseline to four weeks in scores of: 1) depression and anxiety symptoms; 2) subjective evaluations of depressive symptoms, perceived stress, quality of life, aggression, sleep, and pain; and 3) salivary cortisol at eight different timepoints during an ordinary day. Assessments are undertaken by assessors blinded to the randomization group.</p> <p>Trial registration</p> <p>Local Ethics Committee: H-KF 307413</p> <p>Danish Medicines Agency: 2612-3162.</p> <p>EudraCT: 2006-001750-28.</p> <p>Danish Data Agency: 2006-41-6737.</p> <p>ClinicalTrials.gov: NCT 00386841</p