Polytope Expansion of Lie Characters and Applications

The weight systems of finite-dimensional representations of complex, simple Lie algebras exhibit patterns beyond Weyl-group symmetry. These patterns occur because weight systems can be decomposed into lattice polytopes in a natural way. Since lattice polytopes are relatively simple, this decomposition is useful, in addition to being more economical than the decomposition into single weights. An expansion of characters into polytope sums follows from the polytope decomposition of weight systems. We study this polytope expansion here. A new, general formula is given for the polytope sums involved. The combinatorics of the polytope expansion are analyzed; we point out that they are reduced from those of the Weyl character formula (described by the Kostant partition function) in an optimal way. We also show that the weight multiplicities can be found easily from the polytope multiplicities, indicating explicitly the equivalence of the two descriptions. Finally, we demonstrate the utility of the polytope expansion by showing how polytope multiplicities can be used in the calculation of tensor product decompositions, and subalgebra branching rules.Comment: 16 pages incl. 3 figures, to appear in J. Math. Phy

Purely affine elementary su(N) fusions

We consider three-point couplings in simple Lie algebras -- singlets in triple tensor products of their integrable highest weight representations. A coupling can be expressed as a linear combination of products of finitely many elementary couplings. This carries over to affine fusion, the fusion of Wess-Zumino-Witten conformal field theories, where the expressions are in terms of elementary fusions. In the case of su(4) it has been observed that there is a purely affine elementary fusion, i.e., an elementary fusion that is not an elementary coupling. In this note we show by construction that there is at least one purely affine elementary fusion associated to every su(N>3).Comment: 9 pages, LaTe

Field Identification in Non-Unitary Diagonal Cosets

We study the nonunitary diagonal cosets constructed from admissible representations of Kac-Moody algebras at fractional level, with an emphasis on the question of field identification. Generic classes of field identifications are obtained from the analysis of the modular S matrix. These include the usual class related to outer automorphisms, as well as some intrinsically nonunitary field identifications. They allow for a simple choice of coset field representatives where all field components of the coset are associated with integrable finite weights.Comment: 34 page

New Pre-Catalysts for Olefin Polymerisation and Ring-Opening Polymerisation of Lactides/Lactones

A number of pre-catalysts featuring phenoxyimine, phenolate and calixarene ligand systems have been synthesized, characterised and screened for their ability to polymerize α-olefins or ring open polymerize (ROP) cyclic esters. The treatment of a number of related phenoxyimine ligands (L1H – L3H, L6H – L14H) and bridged phenoxyimines (L4H2 and L5H2) with vanadium trichloride (VCl3), vanadium oxytrichloride (VOCl3) or vanadium oxytripropoxide (VO(OnPr)3) afforded the compounds 1 – 20. Compounds 21, 22 and 23 were isolated from the reaction between VO(OnPr)3 and diphenolate ligand (L15H2) or bridged diphenolates (L16H4 or L17H4) respectively. Treatment of the alkali vanadium compound (LiVO(OtBu)4) with tert-butylcalix[6]arene (L18H6) afforded compound 24, while on a number of syntheses, the minor 1D polymeric compound 25 was also obtained. Whilst the reaction between tert-butylcalix[8]arene (L19H8) and NaVO(OtBu)4 led to the formation of compound 26, the use of the alkali free VO(OtBu)3 resulted in the formation of two solvates of compound 27. Imidazole (L20H), oxazole (L21H), α-diimine (L22), iminopyridine (L23, L24) and phenoxyimine (L25H – L29H) ligand sets have been treated with group 5 (Nb or Ta) chlorides or oxytrichlorides to afford compounds 28 – 50, which have been fully characterised. The screening of group 5 compounds 1 – 50 for the polymerisation of α-olefins revealed high activity, significantly in the case of the niobium precatalysts which were two orders of magnitude above the previously reported compounds. The reactions between 1,3-dipropoxy-p-tert-butyl-calix[4]arene (L30H2), hexahomotrioxacalix[3]arene (L31H3) or tripropoxy-p-tert-butylcalix[4]arene (L32H) with zinc or magnesium alkyls has been explored, resulting in the isolation of compounds 51 – 58, which includes a number of heterobimetallic compounds. While all of the zinc and magnesium compounds screened were found to be active for the ring opening polymerisation of either ε-caprolactone or rac-lactide, compound 58, featuring a tripropoxy-p-tert-butylcalix[4]arene (L32H) ligand with a magnesium n-butyl group was found to exhibit exceptional activity and immortal character