Representations of Hopf Ore extensions of group algebras and pointed Hopf algebras of rank one

In this paper, we study the representation theory of Hopf-Ore extensions of group algebras and pointed Hopf algebras of rank one over an arbitrary field $k$. Let H=kG(\chi, a,\d) be a Hopf-Ore extension of $kG$ and $H'$ a rank one quotient Hopf algebra of $H$, where $k$ is a field, $G$ is a group, $a$ is a central element of $G$ and $\chi$ is a $k$-valued character for $G$ with $\chi(a)\neq 1$. We first show that the simple weight modules over $H$ and $H'$ are finite dimensional. Then we describe the structures of all simple weight modules over $H$ and $H'$, and classify them. We also consider the decomposition of the tensor product of two simple weight modules over $H'$ into the direct sum of indecomposable modules. Furthermore, we describe the structures of finite dimensional indecomposable weight modules over $H$ and $H'$, and classify them. Finally, when $\chi(a)$ is a primitive $n$-th root of unity for some $n>2$, we determine all finite dimensional indecomposable projective objects in the category of weight modules over $H'$.Comment: arXiv admin note: substantial text overlap with arXiv:1206.394

Multi-trajectories of triglyceride-glucose index and lifestyle with cardiovascular disease: A cohort study

Background: Previous studies using trajectory models focused on examining the longitudinal changes in triglyceride-glucose (TyG) levels and lifestyle scores separately, without exploring the joint evolution of these two factors. This study aimed to identify the multi-trajectories of TyG levels and lifestyle scores and assess their association with the risk of cardiovascular disease (CVD). Methods: The study enrolled 47,384 participants from three health surveys of the Kailuan Study. The TyG index was computed as Ln [fasting triglycerides (mg/dL) × fasting blood glucose (mg/dL)/2], and the lifestyle scores were derived from five factors, including smoking, alcohol consumption, physical activity, sedentary behaviors, and salt intake. A group-based multi-trajectory model was adopted to identify multi-trajectories of TyG levels and lifestyle scores. The association of identified multi-trajectories with incident CVD was examined using Cox proportional hazard model. Results: Five distinct multi-trajectories of TyG levels and lifestyle scores were identified. During a median follow-up period of 10.98 years, 3042 participants developed CVD events (2481 strokes, 616 myocardial infarctions, and 55 co-current stroke and myocardial infarctions). In comparison to group 3 with the lowest TyG levels and the best lifestyle scores, the highest CVD risk was observed in group 5 characterized by the highest TyG levels and moderate lifestyle scores (HR = 1.76, 95% CI: 1.50–2.05). Group 2 with higher TyG levels and the poorest lifestyle scores had a 1.45-fold (95% CI 1.26–1.66) risk of CVD, and group 1 with lower TyG levels and poorer lifestyle scores had a 1.33-fold (95% CI 1.17–1.50) risk of CVD. Group 4, with moderate TyG levels and better lifestyle scores, exhibited the lowest CVD risk (HR = 1.32, 95% CI: 1.18–1.47). Conclusions: Distinct multi-trajectories of TyG levels and lifestyle scores corresponded to differing CVD risks. The CVD risk caused by a high level TyG trajectory remained increased despite adopting healthier lifestyles. These findings underscored the significance of evaluating the combined TyG and lifestyle patterns longitudinally, and implementing early interventions to reduce CVD risk by lowering TyG levels