A terminal molybdenum carbide prepared by methylidyne deprotonation

The carbide anion [CMo{N(R)Ar}_3]– [R = C(CD_3)_2CH_3, Ar = C_6H_3Me_2-3,5], is obtained by deprotonation of the corresponding methylidyne compound, [HCMo{N(R)Ar}_3], and is characterized by X-ray diffraction as its {K(benzo-15-crown-5)_2}+ salt, thereby providing precedent for the carbon atom as a terminal substituent in transition-metal chemistry

Fusion bases as facets of polytopes

A new way of constructing fusion bases (i.e., the set of inequalities governing fusion rules) out of fusion elementary couplings is presented. It relies on a polytope reinterpretation of the problem: the elementary couplings are associated to the vertices of the polytope while the inequalities defining the fusion basis are the facets. The symmetry group of the polytope associated to the lowest rank affine Lie algebras is found; it has order 24 for \su(2), 432 for \su(3) and quite surprisingly, it reduces to 36 for \su(4), while it is only of order 4 for \sp(4). This drastic reduction in the order of the symmetry group as the algebra gets more complicated is rooted in the presence of many linear relations between the elementary couplings that break most of the potential symmetries. For \su(2) and \su(3), it is shown that the fusion-basis defining inequalities can be generated from few (1 and 2 respectively) elementary ones. For \su(3), new symmetries of the fusion coefficients are found.Comment: Harvmac, 31 pages; typos corrected, symmetry analysis made more precise, conclusion expanded, and references adde

Lie Superalgebras and the Multiplet Structure of the Genetic Code II: Branching Schemes

Continuing our attempt to explain the degeneracy of the genetic code using basic classical Lie superalgebras, we present the branching schemes for the typical codon representations (typical 64-dimensional irreducible representations) of basic classical Lie superalgebras and find three schemes that do reproduce the degeneracies of the standard code, based on the orthosymplectic algebra osp(5|2) and differing only in details of the symmetry breaking pattern during the last step.Comment: 34 pages, 9 tables, LaTe

Generating-function method for fusion rules

This is the second of two articles devoted to an exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper focuses on fusion rules, using the machinery developed for tensor products in the companion article. Although the Kac-Walton algorithm provides a method for constructing a fusion generating function from the corresponding tensor-product generating function, we describe a more powerful approach which starts by first defining the set of fusion elementary couplings from a natural extension of the set of tensor-product elementary couplings. A set of inequalities involving the level are derived from this set using Farkas' lemma. These inequalities, taken in conjunction with the inequalities defining the tensor products, define what we call the fusion basis. Given this basis, the machinery of our previous paper may be applied to construct the fusion generating function. New generating functions for sp(4) and su(4), together with a closed form expression for their threshold levels are presented.Comment: Harvmac (b mode : 47 p) and Pictex; to appear in J. Math. Phy

Generating-function method for tensor products

This is the first of two articles devoted to a exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper is entirely devoted to the study of the tensor-product (infinite-level) limit of fusions rules. We start by reviewing Sharp's character method. An alternative approach to the construction of tensor-product generating functions is then presented which overcomes most of the technical difficulties associated with the character method. It is based on the reformulation of the problem of calculating tensor products in terms of the solution of a set of linear and homogeneous Diophantine equations whose elementary solutions represent ``elementary couplings''. Grobner bases provide a tool for generating the complete set of relations between elementary couplings and, most importantly, as an algorithm for specifying a complete, compatible set of ``forbidden couplings''.Comment: Harvmac (b mode : 39 p) and Pictex; this is a substantially reduced version of hep-th/9811113 (with new title); to appear in J. Math. Phy

Radical anionic versus neutral 2,2′-bipyridyl coordination in uranium complexes supported by amide and ketimide ligands

The synthesis and characterization of (bipy)₂U(N[t-Bu]Ar)₂ (1-(bipy)₂, bipy = 2,2′-bipyridyl, Ar = 3,5-C₆H₃Me₂), (bipy)U(N[1Ad]Ar)₃ (2-bipy), (bipy)₂U(NC[t-Bu]Mes)₃ (3-(bipy)2, Mes = 2,4,6-C₆H₂Me₃), and IU(bipy)(NC[t-Bu]Mes)₃ (3-I-bipy) are reported. X-ray crystallography studies indicate that bipy coordinates as a radical anion in 1-(bipy)₂ and 2-bipy, and as a neutral ligand in 3-I-bipy. In 3-(bipy)₂, one of the bipy ligands is best viewed as a radical anion, the other as a neutral ligand. The electronic structure assignments are supported by NMR spectroscopy studies of exchange experiments with 4,4′-dimethyl-2,2′-bipyridyl and also by optical spectroscopy. In all complexes, uranium was assigned a +4 formal oxidation state.National Science Foundation (U.S.) (Grant CHE-9988806