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In this paper, we study two Z -graded infinite Lie conformal algebras, which are closely related to a class of Lie algebras of the generalized Block type, and which both have a quotient algebra isomorphic to the Virasoro conformal algebra. We concretely determine their isomorphic mappings, conformal derivations, extensions by a one-dimensional center under some conditions, finite conformal modules and Z -graded free intermediate series modules. [ABSTRACT FROM AUTHOR]

In this paper, all Lie bialgebra structures on the derivation Lie algebra W over a rank d ≥ 3 quantum torus associated to q are considered, where q is a d × d matrix with all the entries being roots of unity. They are shown to be triangular coboundary. As a byproduct, it is also proved that the first cohomology group H1 (W, W ⊗ W) is trivial. [ABSTRACT FROM AUTHOR]

We study two classes of non-weight modules Ω (λ , α , β) ⊗ Ind (M) and M (V , Ω (λ , α , β)) over the twisted Heisenberg–Virasoro algebra in this paper. We present conditions under which these modules are irreducible, the necessary and sufficient conditions under which modules in each class are isomorphic, and also show that the irreducible modules in these two classes are new. Finally, we construct non-weight modules Ind y ̲ , λ (C RS) and Ind z ̲ , λ (C PQ) over the twisted Heisenberg–Virasoro algebra and then apply the established results to give irreducible conditions for these modules. [ABSTRACT FROM AUTHOR]

Spindle-like CaCO3 crystals with controllable sizes for papermaking were successfully prepared using CO2 (8% CO2/N2 mixture gas) and calcium carbide residue (CCR) waste, a by-product of acetylene gas and polyvinyl chloride production, as the raw materials by an atomization method at room temperature. The influences of solution concentration, reaction temperature, and gas/liquid flow rate ratios on the properties of the CaCO3 crystal were systematically investigated, and a possible atomization mechanism was proposed. The size of the as-prepared CaCO3 crystal with pure calcite phase was turned from 4.71×4.02 µm to 1.82×1.12 µm by adjusting the reaction conditions. The application of the as-prepared CaCO3 crystals from CCR waste as a filler for papermaking was explored. The R475 blue light whiteness of paper was increased from 77.3 to 80.6 with 11.4% CaCO3 crystals. [ABSTRACT FROM AUTHOR]

The first cohomology group of generalized loop Virasoro algebras with coefficients in the tensor product of its adjoint module is shown to be trivial. The result is used to prove that Lie bialgebra structures on generalized loop Virasoro algebras are coboundary triangular. The authors generalize the results to generalized map Virasoro algebras. [ABSTRACT FROM AUTHOR]

For any complex parameters a and b, W( a, b) is the Lie algebra with basis { L, W | i ∈ ℤ} and relations [ L, L] = ( j − i) L+ j, [ L, W] = ( a + j + bi) W, [ W, W] = 0. In this paper, indecomposable modules of the intermediate series over W( a, b) are classified. It is also proved that an irreducible Harish-Chandra W( a, b)-module is either a highest/lowest weight module or a uniformly bounded module. Furthermore, if a ∋ ℚ, an irreducible weight W( a, b)-module is simply a Vir-module with trivial actions of W's. [ABSTRACT FROM AUTHOR]