YANGIANS AND CLASSICAL LIE ALGEBRAS


Dr Maxim Nazarov, University of York


The aim of the project was to develop further the representation theory of quantized affine Lie algebras and affine Hecke algebras, then to apply the results to the classical representation theory. Several important results were obtained. We studied the irreducible finite-dimensional representations of the Yangian of the Lie algebra glN . The simplest of these representations are parametrized by pairs consisting of a glN -highest weight and of a complex parameter. We obtained an irreducibility criterion for the tensor products of these representations where each of the highest weights is a multiple of a fundamental weight. This generalized recent results of Akasaka-Kashiwara and a well known irreducibility criterion of Zelevinsky. We also obtained a new interpretation of the classical Littlewood-Richardson rule. The grant was used to employ Dr AR Jones as research assistant. He carried out computer experiments necessary for the above described work. Together with him we obtained q-analogues of the projective Young symmetrizers for the symmetric group Sn constructed earler by myself. To achieve our aim we introduced a proper analogue of the affine Hecke algebra due to Iwahori-Matsumoto. Dr Jones also obtained new results on the structure of the projective Young symmetrizers for Sn . By studying the universal R-matrix for the Yangian of glN due to Drinfeld, we obtained a remarkable generalization of the classical Capelli identity (1890). For each polynomial glN -highest weight we gave a certain identity in the ring of differential operators on the tensor product of the N-dimensional and the M-dimensional complex vector spaces with arbitrary M . The Capelli identity corresponds to a fundamental highest weight. We also extended these results from the Lie algebra glN to the Lie superalgebra qN . Our results are available on the e-print archive Mathematics, see


EPSRC Grant and Duration: 59058 over 24 months from 1.3.1996, reference GR/K79406


For further information please contact: Dr Maxim Nazarov, Department of Mathematics, University of York, York YO10 5DD, England; maxim.nazarov@york.ac.uk