Cohomology of osp(1∣2)\frak {osp}(1|2) acting on the space of bilinear differential operators on the superspace R1∣1\mathbb{R}^{1|1}

We compute the first cohomology of the ortosymplectic Lie superalgebra $\mathfrak{osp}(1|2)$ on the (1,1)-dimensional real superspace with coefficients in the superspace $\frak{D}_{\lambda,\nu;\mu}$ of bilinear differential operators acting on weighted densities. This work is the simplest superization of a result by Bouarroudj [Cohomology of the vector fields Lie algebras on $\mathbb{R}\mathbb{P}^1$ acting on bilinear differential operators, International Journal of Geometric Methods in Modern Physics (2005), {\bf 2}; N 1, 23-40]

Cohomology of the Lie Superalgebra of Contact Vector Fields on R1∣1\mathbb{R}^{1|1} and Deformations of the Superspace of Symbols

Following Feigin and Fuchs, we compute the first cohomology of the Lie superalgebra $\mathcal{K}(1)$ of contact vector fields on the (1,1)-dimensional real superspace with coefficients in the superspace of linear differential operators acting on the superspaces of weighted densities. We also compute the same, but $\mathfrak{osp}(1|2)$-relative, cohomology. We explicitly give 1-cocycles spanning these cohomology. We classify generic formal $\mathfrak{osp}(1|2)$-trivial deformations of the $\mathcal{K}(1)$-module structure on the superspaces of symbols of differential operators. We prove that any generic formal $\mathfrak{osp}(1|2)$-trivial deformation of this $\mathcal{K}(1)$-module is equivalent to a polynomial one of degree $\leq4$. This work is the simplest superization of a result by Bouarroudj [On $\mathfrak{sl}$(2)-relative cohomology of the Lie algebra of vector fields and differential operators, J. Nonlinear Math. Phys., no.1, (2007), 112--127]. Further superizations correspond to $\mathfrak{osp}(N|2)$-relative cohomology of the Lie superalgebras of contact vector fields on $1|N$-dimensional superspace

Formal deformations and extensions of `twisted' Lie algebras

The interplay between derivations and algebraic structures has been a subject of significant interest and exploration. Inspired by Yau's twist and the Leibniz rule, we investigate the formal deformation of twisted Lie algebras by invertible derivations, herein referred to as "InvDer Lie". We define representations of InvDer Lie, elucidate cohomology structures of order 1 and 2, and identify infinitesimals as 2-cocycles. Furthermore, we explore central extensions of InvDer Lie, revealing their intricate relationship with cohomology theory.Comment: 1

Two Jahn–Teller systems involved in different kinds of crystal-to-crystal transformations

Two molecular-crystal solids with Jahn-Teller active Cu(II) centers undergo crystal-to-crystal transformations by different routes. The first example is a coordination copolymer with alternating Co and Cu centers along the polymer chain, and with a charge of (+1) for each link in the chain. Charge is balanced by an anion whose composition is identical to the Cocentered link of the copolymer. The overall composition can be described as {[Co(orot)2(bpy)][μCu(bpy)(H2O)]}n[Co(orot)2(bpy)]n‧5nH2O, 1, in which H2orot is orotic acid, C5H4N2O4. With gentle heating in dry nitrogen gas, crystals of this compound undergo a chemical reaction in which the anion is incorporated into the polymer as a metalloligand with one oxygen atom of the original anion substituting an aqua ligand on the Cu center of the polymer. Structure analysis at intermediate stages of the process indicate that substitution occurs by an associative mechanism. The second example involves a Jahn-Teller intermediate, formed in solution and isolable as a crystalline precipitate, which when left in contact with the reaction mixture undergoes a solvent mediated crystal-to-crystal transformation in which the two axial ligands involved in Jahn-Teller elongation are lost. The intermediate Cs2[transCu(orot)2(H2O)2]∙4H2O, 2, proceeds to the simple square-planar final product, Cs2[trans-Cu(orot)2]∙3H2O, 3. It is noted that the nickel-centered analogue of compound 2 does not undergo further transformation to a square-planar product

A separation principle for the stabilisation of a class of fractional order time delay nonlinear systems

DOI: 10.1017/S000497271800083