Generalized P(N)-graded Lie superalgebras

We generalize the P(N)-graded Lie superalgebras of Martinez-Zelmanov. This generalization is not so restrictive but suffcient enough so that we are able to have a classification for this generalized P(N)-graded Lie superalgebras. Our result is that the generalized P(N)-graded Lie super-algebra L is centrally isogenous to a matrix Lie superalgebra coordinated by an associative superalgebra with a super-involution. Moreover, L is P(N)-graded if and only if the coordinate algebra R is commutative and the super-involution is trivial. This recovers Martinez-Zelmanov's theorem for type P(N). We also obtain a generalization of Kac's coordinatization via Tits-Kantor-Koecher construction. Actually, the motivation of this generalization comes from the Fermionic-Bosonic module construction.     

算子代数上2-局部Lie三重导子的结构

设X是维数大于2的Banach空间,映射δ:B(X)→B(X)是2-局部Lie三重导子,则对所有A∈B(X)有δ(A)=[A,T]+φ(A),这里T∈B(X),φ是从B(X)到FI的齐次映射且满足对所有A,B∈B(X)有φ(A+B)=φ(A),其中B是交换子的和.     

Quantizer–dequantizer operators as a tool for formulating the quantization procedure

We consider the quantization procedure and investigate the application of the quantizer–dequantizer method and star-product technique to construct associative products and the associative algebras formed by the quantizer–dequantizer operators and their symbols. The corresponding Lie algebras are also constructed. We study the case where the quantizer–dequantizer operators form a self-dual system and show that the structure constants of the Lie algebras satisfy some identity, in addition to the Jacobi identity. Using tomographic quantizer–dequantizer operators and their symbols, we construct the continuous associative algebra and the corresponding Lie algebra.     

Modular Synthetic Approach to Carboranyl‒Biomolecules Conjugates

The development of novel, tumor-selective and boron-rich compounds as potential agents for use in boron neutron capture therapy (BNCT) represents a very important field in cancer treatment by radiation therapy. Here, we report the design and synthesis of two promising compounds that combine meta-carborane, a water-soluble monosaccharide and a linking unit, namely glycine or ethylenediamine, for facile coupling with various tumor-selective biomolecules bearing a free amino or carboxylic acid group. In this work, coupling experiments with two selected biomolecules, a coumarin derivative and folic acid, were included. The task of every component in this approach was carefully chosen: the carborane moiety supplies ten boron atoms, which is a tenfold increase in boron content compared to the l-boronophenylalanine (l-BPA) presently used in BNCT; the sugar moiety compensates for the hydrophobic character of the carborane; the linking unit, depending on the chosen biomolecule, acts as the connection between the tumor-selective component and the boron-rich moiety; and the respective tumor-selective biomolecule provides the necessary selectivity. This approach makes it possible to develop a modular and feasible strategy for the synthesis of readily obtainable boron-rich agents with optimized properties for potential applications in BNCT.     

Synthesis and characterization of modular polyphosphonate homopolymers and copolymers

Linear polyphosphonates with the generic formula –[P(Ph)(X)OR′O]_n– (X = S or Se) have been synthesized by polycondensations of P(Ph)(NEt₂)₂ and a diol (HOR′OH = 1,4-cyclohexanedimethanol, 1,4-benzenedimethanol, tetraethylene glycol, or 1,12-dodecanediol) followed by reaction with a chalcogen. Random copolymers have been synthesized by polycondensations of P(Ph)(NEt₂)₂ and mixture of two of the diols in a 2:1:1 mol ratio followed by reaction with a chalcogen. Block copolymers with the generic formula –[P(Ph)(X)OR′O]_{(x + 2)} –[P(Ph)(X)OR′O]_{(x + 3)}– (X = S or Se) have been synthesized by the polycondensations of Et₂N[P(Ph)(X)OR′O]_{(x + 2)}P(Ph)NEt₂ oligomers with HOR′O[P(Ph)(X)OR′O]_{(x + 3)}H oligomers followed by reaction with a chalcogen. The Et₂N[P(Ph)(X)OR′O]_{(x + 2)}P(Ph)NEt₂ oligomers are prepared by the reaction of an excess of P(Ph)(NEt₂)₂ with a diol while the HOR′O[P(Ph)(X)OR′O]_{(x + 3)}H oligomers are prepared by the reaction of P(Ph)(NEt₂)₂ with an excess of the diol. In each case the excess, x is the same and determines the average block sizes. All of the polymers were characterized using ¹H, ¹³C{¹H}, and ³¹P{¹H} NMR spectroscopy, TGA, DSC, and SEC. ³¹P{¹H} NMR spectroscopy demonstrates that the random and block copolymers have the expected arrangements of monomers and, in the case of block copolymers, verifies the block sizes. All polymers are thermally stable up to ~300°C, and the arrangements of monomers in the copolymers (block vs. random) affect their degradation temperatures and T_g profiles. The polymers have weight average MWs of up to 3.8 × 10⁴ Da.     

The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair consisting of a left‐invariant Riemannian metric g and a positive constant c such that , where is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that is solvable for some left‐invariant Riemannian metric g.