Conserved quantities are crucial in quantum physics. Here we discuss a general scenario of Hamiltonians. All the Hamiltonians within this scenario share a common conserved quantity form. For unitary parametrization processes, the characteristic opera
tor of this scenario is analytically provided, as well as the corresponding quantum Fisher information (QFI). As the application of this scenario, we focus on two classes of Hamiltonians: su(2) category and canonical category. Several specific physical systems in these two categories are discussed in detail. Besides, we also calculate an alternative form of QFI in this scenario.
To classify the finite dimensional pointed Hopf algebras with $G= {rm HS}$ or ${rm Co3}$ we obtain the representatives of conjugacy classes of $G$ and all character tables of centralizers of these representatives by means of software {rm GAP}.
Based on a cosmological N-body simulation we analyze spatial and kinematic alignments of satellite halos within six times the virial radius of group size host halos (Rvir). We measure three different types of spatial alignment: halo alignment between
the orientation of the group central substructure (GCS) and the distribution of its satellites, radial alignment between the orientation of a satellite and the direction towards its GCS, and direct alignment between the orientation of the GCS and that of its satellites. In analogy we use the directions of satellite velocities and probe three further types of alignment: the radial velocity alignment between the satellite velocity and connecting line between satellite and GCS, the halo velocity alignment between the orientation of the GCS and satellite velocities and the auto velocity alignment between the satellites orientations and their velocities. We find that satellites are preferentially located along the major axis of the GCS within at least 6 Rvir (the range probed here). Furthermore, satellites preferentially point towards the GCS. The most pronounced signal is detected on small scales but a detectable signal extends out to 6 Rvir. The direct alignment signal is weaker, however a systematic trend is visible at distances < 2 Rvir. All velocity alignments are highly significant on small scales. Our results suggest that the halo alignment reflects the filamentary large scale structure which extends far beyond the virial radii of the groups. In contrast, the main contribution to the radial alignment arises from the adjustment of the satellite orientations in the group tidal field. The projected data reveal good agreement with recent results derived from large galaxy surveys. (abridged)
