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

Annihilation Rates of Heavy 1−−1^{--} S-wave Quarkonia in Salpeter Method

The annihilation rates of vector $1^{--}$ charmonium and bottomonium $^3S_1$ states $V \rightarrow e^+e^-$ and $V\rightarrow 3\gamma$, $V \rightarrow \gamma gg$ and $V \rightarrow 3g$ are estimated in the relativistic Salpeter method. We obtained $\Gamma(J/\psi\rightarrow 3\gamma)=6.8\times 10^{-4}$ keV, $\Gamma(\psi(2S)\rightarrow 3\gamma)=2.5\times 10^{-4}$ keV, $\Gamma(\psi(3S)\rightarrow 3\gamma)=1.7\times 10^{-4}$ keV, $\Gamma(\Upsilon(1S)\rightarrow 3\gamma)=1.5\times 10^{-5}$ keV, $\Gamma(\Upsilon(2S)\rightarrow 3\gamma)=5.7\times 10^{-6}$ keV, $\Gamma(\Upsilon(3S)\rightarrow 3\gamma)=3.5\times 10^{-6}$ keV and $\Gamma(\Upsilon(4S)\rightarrow 3\gamma)=2.6\times 10^{-6}$ keV. In our calculations, special attention is paid to the relativistic correction, which is important and can not be ignored for excited $2S$, $3S$ and higher excited states.Comment: 10 pages,2 figures, 5 table

Separation of Different Contributions to the Total X-ray Luminosity in Gamma-ray Loud Blazars

The relativistic beaming model has been successfully used to explain many of the observational properties of active galactic nuclei. In this model the total emission is formed by two components, one beamed, one unbeamed. However, the exact contribution from each component in unresolved sources is still not clear. In the radio band, the core and extended emissions are clearly separated. We adopt the method proposed by Kembhavi to separate the two contributions in the X-ray emissions in a sample of 19 gamma-ray loud blazars. It is clearly shown that the beamed emission dominates the X-ray flux and the unbeamed X-ray emission is correlated with the extended radio emission of the considered objects. We also find that the ratio of the beamed to the unbeamed X-ray luminosity is correlated with the X-ray spectral index, an effect that should be a consequence of the underlying X-ray emission mechanism.Fil: Fan, Jun Hui. Guangzhou University. Center for Astrophysics; ChinaFil: Romero, Gustavo Esteban. Provincia de Buenos Aires. Gobernación. Comisión de Investigaciones Científicas. Instituto Argentino de Radioastronomía. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto Argentino de Radioastronomía; ArgentinaFil: Wang, Yong Xiang. College of Science and Trade; ChinaFil: Zhang, Jiang Shui. Guangzhou University. Center for Astrophysics; Chin