Relative category O, blocks, and representation type

Let A be a finite dimensional algebra over a field k. We can place A into one of three classes, according to the indecomposable modules the algebra admits. The algebra has finite representation type if it has only finitely many indecomposable modules, up to isomorphism

Cohomology and Support Varieties for Lie Superalgebras II

In \cite{BKN} the authors initiated a study of the representation theory of classical Lie superalgebras via a cohomological approach. Detecting subalgebras were constructed and a theory of support varieties was developed. The dimension of a detecting subalgebra coincides with the defect of the Lie superalgebra and the dimension of the support variety for a simple supermodule was conjectured to equal the atypicality of the supermodule. In this paper the authors compute the support varieties for Kac supermodules for Type I Lie superalgebras and the simple supermodules for $\mathfrak{gl}(m|n)$. The latter result verifies our earlier conjecture for $\mathfrak{gl}(m|n)$. In our investigation we also delineate several of the major differences between Type I versus Type II classical Lie superalgebras. Finally, the connection between atypicality, defect and superdimension is made more precise by using the theory of support varieties and representations of Clifford superalgebras.Comment: 28 pages, the proof of Proposition 4.5.1 was corrected, several other small errors were fixe

Complexity for Modules Over the Classical Lie Superalgebra gl(m|n)

Let $\mathfrak{g}=\mathfrak{g}_{\bar{0}}\oplus \mathfrak{g}_{\bar{1}}$ be a classical Lie superalgebra and $\mathcal{F}$ be the category of finite dimensional $\mathfrak{g}$-supermodules which are completely reducible over the reductive Lie algebra $\mathfrak{g}_{\bar{0}}$. In an earlier paper the authors demonstrated that for any module $M$ in $\mathcal{F}$ the rate of growth of the minimal projective resolution (i.e., the complexity of $M$) is bounded by the dimension of $\mathfrak{g}_{\bar{1}}$. In this paper we compute the complexity of the simple modules and the Kac modules for the Lie superalgebra $\mathfrak{gl}(m|n)$. In both cases we show that the complexity is related to the atypicality of the block containing the module.Comment: 32 page

Temporal relationship between instantaneous pressure gradients and peak‐to‐peak systolic ejection gradient in congenital aortic stenosis

ObjectiveWe sought to identify a time during cardiac ejection when the instantaneous pressure gradient (IPG) correlated best, and near unity, with peak‐to‐peak systolic ejection gradient (PPSG) in patients with congenital aortic stenosis. Noninvasive echocardiographic measurement of IPG has limited correlation with cardiac catheterization measured PPSG across the spectrum of disease severity of congenital aortic stenosis. A major contributor is the observation that these measures are inherently different with a variable relationship dependent on the degree of stenosis.DesignHemodynamic data from cardiac catheterizations utilizing simultaneous pressure measurements from the left ventricle (LV) and ascending aorta (AAo) in patients with congenital valvar aortic stenosis was retrospectively reviewed over the past 5 years. The cardiac cycle was standardized for all patients using the percentage of total LV ejection time (ET). Instantaneous gradient at 5% intervals of ET were compared to PPSG using linear regression and Bland‐Altman analysis.ResultsA total of 22 patients underwent catheterization at a median age of 13.7 years (interquartile range [IQR] 10.3‐18.0) and median weight of 51.1 kg (IQR 34.2‐71.6). The PPSG was 46.5 ± 12.6 mm Hg (mean ± SD) and correlated suboptimally with the maximum and mean IPG. The midsystolic IPG (occurring at 50% of ET) had the strongest correlation with the PPSG (PPSG = 0.97(IPG50%)–1.12, R2 = 0.88), while the IPG at 55% of ET was closest to unity (PPSG = 0.997(IPG55%)–1.17, R2 = 0.87).ConclusionsThe commonly measured maximum and mean IPG are suboptimal estimates of the PPSG in congenital aortic stenosis. Using catheter‐based data, IPG at 50%–55% of ejection correlates well with PPSG. This may allow for a more accurate estimation of PPSG via noninvasive assessment of IPG.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/140042/1/chd12514.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/140042/2/chd12514_am.pd

Trauma Team Activation for Geriatric Trauma at a Level II Trauma Center: Are the Elderly Under-triaged?

Abstract Geriatric patients often sustain life-threatening injuries from minor trauma. A growing body of research suggests that these patients are often under-triaged in the emergency setting.The purpose of this research was to evaluate whether or not geriatric trauma patients are under-triaged at a community based level II trauma center. 1434 trauma patients over the age of 65 presenting from 2010-2015 were retrospectively reviewed from the Cabell Huntington Hospital trauma registry and analyzed for age, gender, arrival type, ED response, Glasgow Coma Scale (GCS), Injury Severity Score (ISS), injury cause, ICD-9 diagnosis codes, and mortality. Under-triage and over-triage rates were determined using the Cribari method (under-triage = ISS ≥ 16 without full trauma team activation [TTA]; Over-triage = ISS ≤ 15 with full TTA). The under-triage rate was 9.5% (132/1393) with the majority of under-triaged patients having head trauma (n=423). There were 371 head trauma patients with a recorded GCS and analysis shows those with a GCS ≥ 13 had a 1.2% mortality risk (n=326; ISS 10.2), but that risk drastically increases to 60% with GSC ≤ 12 (n=45; ISS 21.5). Of the 45 patients with GSC ≤ 12, only 4% had priority 1 TTA using the current protocol (2/45). The American College of Surgeons-Committee of Trauma (ACS-COT) recommends an acceptable under-triage rate of \u3c 5%. In order to improve geriatric care and reduce under-triage rates, we recommend that an age-based criteria be added to our TTA protocol at our community based Level II trauma center: priority 1 TTA for all patients 65 years or older sustaining head trauma with a GCS ≤ 12 or suspicion of intracranial hemorrhage