On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, Suter.Comment: Latex file, 35 pages, minor corrections, some examples added. To appear in Selecta Mathematic

Root polytopes and abelian ideals

We study the root polytope $\mathcal P_\Phi$ of a finite irreducible crystallographic root system $\Phi$ using its relation with the abelian ideals of a Borel subalgebra of a simple Lie algebra with root system $\Phi$. We determine the hyperplane arrangement corresponding to the faces of codimension 2 of $\mathcal P_\Phi$ and analyze its relation with the facets of $\mathcal P_\Phi$. For $\Phi$ of type $A_n$ or $C_n$, we show that the orbits of some special subsets of abelian ideals under the action of the Weyl group parametrize a triangulation of $\mathcal P_\Phi$. We show that this triangulation restricts to a triangulation of the positive root polytope $\mathcal P_\Phi^+$.Comment: 41 pages, revised version, accepted for publication in Journal of Algebraic Combinatoric

Radiographic volume analysis as a novel tool to determine nasopalatine duct cyst dimensions and its association with presenting symptoms and postoperative complications.

OBJECTIVES: The aims of the study were to use cone beam computed tomography (CBCT) images of nasopalatine duct cysts (NPDC) and to calculate the diameter, surface area, and 3D-volume using a custom-made software program. Furthermore, any associations of dimensions of NPDC with age, gender, presence/absence of maxillary incisors/canines (MI/MC), endodontic treatment of MI/MC, presenting symptoms, and postoperative complications were evaluated. MATERIAL AND METHODS: The study comprised 40 patients with a histopathologically confirmed NPDC. On preoperative CBCT scans, curves delineating the cystic borders were drawn in all planes and the widest diameter (in millimeter), surface area (in square millimeter), and volume (in cubic millimeter) were calculated. RESULTS: The overall mean cyst diameter was 15 mm (range 7-47 mm), the mean cyst surface area 566 mm(2) (84-4,516 mm(2)), and the mean cyst volume 1,735 mm(3) (65-25,350 mm(3)). For 22 randomly allocated cases, a second measurement resulted in a mean absolute aberration of ±4.2 % for the volume, ±2.8 % for the surface, and ±4.9 % for the diameter. A statistically significant association was found for the CBCT determined cyst measurements and the need for preoperative endodontic treatment to MI/MC and for postoperative complications. CONCLUSION: In the hands of a single experienced operator, the novel software exhibited high repeatability for measurements of cyst dimensions. Further studies are needed to assess the application of this tool for dimensional analysis of different jaw cysts and lesions including treatment planning. CLINICAL RELEVANCE: Accurate radiographic information of the bone volume lost (osteolysis) due to expansion of a cystic lesion in three dimensions could help in personalized treatment planning