### YANGIANS AND CLASSICAL LIE ALGEBRAS

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 *gl*_{N} .
The simplest of these representations are parametrized by pairs consisting
of a *gl*_{N} -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
*S*_{n}
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 *S*_{n} .
By studying the universal *R*-matrix for the Yangian of
*gl*_{N}
due to Drinfeld, we obtained a remarkable generalization of the classical
Capelli identity (1890). For each polynomial
*gl*_{N} -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 *gl*_{N}
to the Lie superalgebra *q*_{N} .
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`