Derived HH-module endomorphism rings

Let $H$ be a Hopf algebra, $A/B$ be an $H$-Galois extension. Let $D(A)$ and $D(B)$ be the derived categories of right $A$-modules and of right $B$-modules respectively. An object $M^\cdot\in D(A)$ may be regarded as an object in $D(B)$ via the restriction functor. We discuss the relations of the derived endomorphism rings E_A(M^\cdot)=\op_{i\in\mathbb{Z}}\Hom_{D(A)}(M^\cdot,M^\cdot[i]) and E_B(M^\cdot)=\op_{i\in\mathbb{Z}}\Hom_{D(B)}(M^\cdot,M^\cdot[i]). If $H$ is a finite dimensional semisimple Hopf algebra, then $E_A(M^\cdot)$ is a graded subalgebra of $E_B(M^\cdot)$. In particular, if $M$ is a usual $A$-module, a necessary and sufficient condition for $E_B(M)$ to be an $H^*$-Galois graded extension of $E_A(M)$ is obtained. As an application of the results, we show that the Koszul property is preserved under Hopf Galois graded extensions.Comment: to appear at Glasgow Mathematical Journa

Hopf algebra actions on differential graded algebras and applications

Let $H$ be a finite dimensional semisimple Hopf algebra, $A$ a differential graded (dg for short) $H$-module algebra. Then the smash product algebra $A\#H$ is a dg algebra. For any dg $A\#H$-module $M$, there is a quasi-isomorphism of dg algebras: \mathrm{RHom}_A(M,M)\#H\longrightarrow \mathrm{RHom}_{A\#H}(M\ot H,M\ot H). This result is applied to $d$-Koszul algebras, Calabi-Yau algebras and AS-Gorenstein dg algebrasComment: to appear in Bull. Belg. Math So

Deformations of Koszul Artin-Schelter Gorenstein algebras

We compute the Nakayama automorphism of a PBW-deformation of a Koszul Artin-Schelter Gorenstein algebra of finite global dimension, and give a criterion for an augmented PBW-deformation of a Koszul Calabi-Yau algebra to be Calabi-Yau. The relations between the Calabi-Yau property of augmented PBW-deformations and that of non-augmented cases are discussed. The Nakayama automorphisms of PBW-deformations of Koszul Artin-Schelter Gorenstein algebras of global dimensions 2 and 3 are given explicitly. We show that if a PBW-deformation of a graded Calabi-Yau algebra is still Calabi-Yau, then it is defined by a potential under some mild conditions. Some classical results are also recovered. Our main method used in this paper is elementary and based on linear algebra. The results obtained in this paper will be applied in a subsequent paper.Comment: to appear at Manuscripta Mat

Unification of Flavor SU(3) Analyses of Heavy Hadron Weak Decays

Analyses of heavy mesons and baryons hadronic charmless decays using the flavor SU(3) symemtry can be formulated in two different forms. One is to construct the SU(3) irreducible representation amplitude (IRA) by decomposing effective Hamiltonian, and the other is to draw the topological diagrams (TDA). In the flavor SU(3) limit, we study various $B/D\to PP,VP,VV$, $B_c\to DP/DV$ decays, and two-body nonleptonic decays of beauty/charm baryons, and demonstrate that when all terms are included these two ways of analyzing the decay amplitudes are completely equivalent. Furthermore we clarify some confusions in drawing topological diagrams using different ways of describing beauty/charm baryons.Comment: 36 pages, 6 figures, 16 table

Three photon absorption in ZnO and ZnS crystals

We report a systematic investigation of both three-photon absorption (3PA)spectra and wavelength dispersions of Kerr-type nonlinear refraction in wide-gap semiconductors. The Z-scan measurements are recorded for both ZnO and ZnS with femtosecond laser pulses. While the wavelength dispersions of the Kerr nonlinearity are in agreement with a two-band model, the wavelength dependences of the 3PA are found to be given by (3Ephoton/Eg-1)5/2(3Ephoton/Eg)-9. We also evaluate higher-order nonlinear optical effects including the fifth-order instantaneous nonlinear refraction associated with virtual three-photon transitions, and effectively seventh-order nonlinear processes induced by three-photon-excited free charge carriers. These higher-order nonlinear effects are insignificant with laser excitation irradiances up to 40 GW/cm2. Both pump-probe measurements and three-photon figures of merits demonstrate that ZnO and ZnS should be a promising candidate for optical switching applications at telecommunication wavelengths.Comment: 13 pages, 7 figure

Gapped spin liquid with Z2\mathbb{Z}_2-topological order for kagome Heisenberg model

We apply symmetric tensor network state (TNS) to study the nearest neighbor spin-1/2 antiferromagnetic Heisenberg model on Kagome lattice. Our method keeps track of the global and gauge symmetries in TNS update procedure and in tensor renormalization group (TRG) calculation. We also introduce a very sensitive probe for the gap of the ground state -- the modular matrices, which can also determine the topological order if the ground state is gapped. We find that the ground state of Heisenberg model on Kagome lattice is a gapped spin liquid with the $\mathbb{Z}_2$-topological order (or toric code type), which has a long correlation length $\xi\sim 10$ unit cell length. We justify that the TRG method can handle very large systems with over thousands of spins. Such a long $\xi$ explains the gapless behaviors observed in simulations on smaller systems with less than 300 spins or shorter than 10 unit cell length. We also discuss experimental implications of the topological excitations encoded in our symmetric tensors.Comment: 10 pages, 7 figure

Theoretical investigation of the thermal performance of a novel solar loop-heat-pipe façade-based heat pump water heating system

The aim of the paper was to present a dedicated theoretical investigation into the thermal performance of a novel solar loop-heat-pipe façade based heat pump water heating system. This involved thermo-fluid analyses, computer numerical model development, the model running up, modelling result analyses and conclusion. An energy balance network was established on each part and the whole range of the system to address the associated energy conversion and transfer processes. On basis of this, a computer numerical model was developed and run up to predict the thermal performance of such a system at different system configurations, layouts and operational conditions. It was suggested that the loop heat pipes could be filled with either water, R134a, R22 or R600a; of which R600a is the favourite working fluid owing to its relatively larger heat transfer capacity and positive pressure in operation. Variations in the system configuration, i.e., glazing covers, heat exchangers, would lead to identifiable differences in the thermal performance of the system, represented by the thermal efficiency and COP. Furthermore, impact of the external operational parameters, i.e., solar radiation and ambient air temperature, to the system's thermal performance was also investigated. The research was based on an innovative loop-heat-pipe façade and came up with useful results reflecting the thermal performance of the combined system between the façade and heat pump. This would help promote development and market penetration of such an innovative solar heating technology, and thus contribute to achieving the global targets in energy saving and carbon emission reduction