On the structure of maximal solvable extensions and of Levi extensions of nilpotent algebras

We establish an improved upper estimate on dimension of any solvable algebra s with its nilradical isomorphic to a given nilpotent Lie algebra n. Next we consider Levi decomposable algebras with a given nilradical n and investigate restrictions on possible Levi factors originating from the structure of characteristic ideals of n. We present a new perspective on Turkowski's classification of Levi decomposable algebras up to dimension 9.Comment: 21 pages; major revision - one section added, another erased; author's version of the published pape

All solvable extensions of a class of nilpotent Lie algebras of dimension n and degree of nilpotency n-1

We construct all solvable Lie algebras with a specific n-dimensional nilradical n_(n,2) (of degree of nilpotency (n-1) and with an (n-2)-dimensional maximal Abelian ideal). We find that for given n such a solvable algebra is unique up to isomorphisms. Using the method of moving frames we construct a basis for the Casimir invariants of the nilradical n_(n,2). We also construct a basis for the generalized Casimir invariants of its solvable extension s_(n+1) consisting entirely of rational functions of the chosen invariants of the nilradical.Comment: 19 pages; added references, changes mainly in introduction and conclusions, typos corrected; submitted to J. Phys. A, version to be publishe

Long-range magnetic ordering in Li2MnGeO4 and precursor short-range spin correlations

Long-range magnetic ordering and short-range spin correlations in layered noncentrosymmetric orthogermanate Li2MnGeO4 were studied by means of polarized and unpolarized neutron scattering. The combined Rietveld refinement of synchrotron and neutron powder diffraction data at room temperature within the Pmn2(1) space group allowed us to specify the details of the crystal structure. According to the additional Bragg peaks in low-temperature neutron diffraction patterns a long-range antiferromagnetic ordering with the propagation vector k = (1/2 1/2 1/2) has been found below T-N approximate to 8 K. Symmetry analysis revealed the model of the ground state spin structure within the C(a)c (no. 9.41) magnetic space group. It is represented by the noncollinear ordering of manganese atoms with a refined magnetic moment of 4.9 mu(B)/Mn2+ at 1.7 K, which corresponds to the saturated value for the high-spin configuration S = 5/2. Diffuse magnetic scattering was detected on the neutron diffraction patterns at temperatures just above T-N. Its temperature evolution was investigated in detail by polarized neutron scattering with the following XYZ-polarization analysis. Reverse Monte Carlo simulation of diffuse scattering data showed the development of short-range ordering in Li2MnGeO4, which is symmetry consistent on a small scale with the long-range magnetic state below T-N. The reconstructed radial spin-pair correlation function S(0)S(r) displayed the predominant role of antiferromagnetic correlations. It was found that spin correlations are significant only for the nearest magnetic neighbors and almost disappear at r approximate to 12 angstrom at 10 K. Temperature dependence of the diffuse scattering implies short-range ordering long before the magnetic phase transition. Besides, the spin arrangement was found to be similar in both cases above and below T-N. As a result, an exhaustive picture of the gradual formation of magnetic ordering in Li2MnGeO4 is presented

Long-range magnetic ordering in Li2_2MnGeO4_4 and precursor short-range spin correlations

Long-range magnetic ordering and short-range spin correlations in layered noncentrosymmetric orthogermanate Li2MnGeO4 were studied by means of polarized and unpolarized neutron scattering. The combined Rietveld refinement of synchrotron and neutron powder diffraction data at room temperature within the Pmn21 space group allowed us to specify the details of the crystal structure. According to the additional Bragg peaks in low-temperature neutron diffraction patterns a long-range antiferromagnetic ordering with the propagation vector k=(1/21/21/2) has been found below TN≈8K. Symmetry analysis revealed the model of the ground state spin structure within the Cac (no. 9.41) magnetic space group. It is represented by the noncollinear ordering of manganese atoms with a refined magnetic moment of 4.9μB/Mn2+ at 1.7 K, which corresponds to the saturated value for the high-spin configuration S=5/2. Diffuse magnetic scattering was detected on the neutron diffraction patterns at temperatures just above TN. Its temperature evolution was investigated in detail by polarized neutron scattering with the following XYZ-polarization analysis. Reverse Monte Carlo simulation of diffuse scattering data showed the development of short-range ordering in Li2MnGeO4, which is symmetry consistent on a small scale with the long-range magnetic state below TN. The reconstructed radial spin-pair correlation function S(0)S(r) displayed the predominant role of antiferromagnetic correlations. It was found that spin correlations are significant only for the nearest magnetic neighbors and almost disappear at r≈12Å at 10 K. Temperature dependence of the diffuse scattering implies short-range ordering long before the magnetic phase transition. Besides, the spin arrangement was found to be similar in both cases above and below TN. As a result, an exhaustive picture of the gradual formation of magnetic ordering in Li2MnGeO4 is presented